I am trying to understand this paper: http://link.aps.org/doi/10.1103/PhysRevLett.99.236809
(Here is an arXiv version: http://arxiv.org/abs/0709.1274)
In the introduction, they mention certain symmetry arguments (the two paragraphs in the second column of the first page). Unfortunately, I am ill-equipped to understand these symmetry arguments. Would it be possible for an expert to walk me through these two paragraphs?
I am sorry if this is a poorly worded question (this is my first post here).
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As per the comments, I am copying the relevant paragraphs here:
“Before starting specific calculations, it will be instructive to make
some general symmetry analysis. A valley contrasting magnetic moment
has the relation $ \mathfrak{m}_v=\chi \tau_z $, where $\tau_z = \pm 1$ labels the
two valleys and $\chi$ is a coefficient characterizing the material.
Under time reversal, $\mathfrak{m}_v$
changes sign, and so does $\tau_z$ (the two valleys switch when the
crystal momentum changes sign). Therefore, $\chi$ can be non-zero even
if the system is non-magnetic. Under spatial inversion, only $\tau_z$ changes sign.
Therefore $\mathfrak{m}_v$ can be nonzero only in systems with broken inversion symmetry.
Inversion symmetry breaking simultaneously allows a valley Hall
effect, with $\mathbf j^v = \sigma^v_H \hat{\mathbf z} \times \mathbf
E$, where $\sigma^v_H$ is the transport coefficient (valley Hall
conductivity), and the valley current $\mathbf j^v$ is defined as the
average of the valley index times the velocity operator. Under time
reversal, both the valley current and electric field are invariant.
Under spatial inversion, the valley current is still invariant but
the electric field changes sign. Therefore, the valley Hall
conductivity can be non-zero when the inversion symmetry is broken,
even if the time reversal symmetry remains.''
Best Answer
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.