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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?

Diagram depicting Zeeman effect in K and -K valley excitations

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

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  • 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.

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  • $\begingroup$ Thank you for the answer. I agree that the valleys for a null magnetic field are connected by time reversal symmetry. My doubt is that in this article they mention "However, in analogy with conventional spin degrees of freedom, this K/K' valley degeneracy can be lifted by external magnetic fields B, which break time-reversal symmetry". This makes me confused, because I don't understand why it says the Zeeman effect breaks the time-reversal symmetry. Did you understand this? $\endgroup$ Jul 25, 2021 at 15:32
  • $\begingroup$ @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. $\endgroup$
    – Jack
    Jul 26, 2021 at 0:06

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