# Valleys and time reversal symmetry (Zeeman effect)

Right now I'm focused on transition metal dichalcogenides. These compounds in the Brillouin zone have valleys in the valence band and in the conduction band at points K and -K. From what I've seen of the Zeeman effect, as you apply a magnetic field, you break the time reversal symmetry. But in this Youtube video close to the 11:15 minute, they mention that as the K and -K valleys are connected by the time reversal symmetry, so the g-factors are going to be 2 and -2, respectively. I don't quite understand why they say that applying a magnetic field in this case breaks the time reverse symmetry, but at the same time they say that the valleys in K and -K are joined by time reversal symmetry. It seems a bit contradictory. Could you explain to me why it is not a contradiction?

(Zeeman effect contributions image taken from Srivastava, Ajit, Sidler, Meinrad, Allain, Adrien, Lembke, Dominik, Kis, Andras, Imamoğlu, A., Valley Zeeman effect in elementary optical excitations of monolayer WSe$$_\mathit{2}$$. Nature Physics, 11:141-147, 2015.)

• First of all, H-phase TMDC monolayers like $$\ce{WSe_2}$$ are non-magnetic semiconducting materials, which means the time-reversal symmetry (TRS) is preserved. Therefore, you can say the energy states at $$K$$ and $$-K$$ are connected by TRS, seeing this post.
• For H-phase TMDC monolayers, the inversion symmetry (IS) is broken. In addition, the spin-orbit coupling (SOC) in these materials is strong. The combination of broken IS and strong SOC leads to the spin splittings at valleys $$K$$/$$-K$$. In fact, the splitting at valleys $$K/-K$$ likes Zeeman-type splitting.
• @CarmenGonzález [The bands and optical transitions at the K and K$_0$ valleys are nominally degenerate in energy and related by time-reversal symmetry. However, in analogy with conventional spin degrees of freedom, this K/K$_0$ valley degeneracy can be lifted by external magnetic fields B, which break time-reversal symmetry]. ==> 1. K/K$_0$ valleys degenerate in energy. 2. Valleys are coupled to spin by time-reversal symmetry. 3. Magnetic fields operate on the spin. Therefore, the authors can break the valley degeneracy with an external magnetic field, this is what they are talking about.