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Suppose the spin orbit coupling term is written below. $$H_{soc} = \xi L\cdot\sigma$$ where, $\xi$ is the strength of the spin orbit coupling, $L$ and $\sigma$ are angular momentum and pauli matrix operators respectively.

The exchange potential term is written in the following. $$m_z \times \sigma$$ where, $m_z$ is a magnitude of exchange potential and $\sigma$ is the pauli matrix

When the time reversal symmetry operator is applied onto these two terms respectively, why does the spin orbit coupling term change the sign; while, does not the exchange potential term?

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    $\begingroup$ Spin orbit does not change sign under time reversal that's the whole point of Kramers theorem. $\endgroup$
    – Mauricio
    Commented Mar 17, 2022 at 9:41
  • $\begingroup$ @Mauricio Thank you for the reply. In other words, if I multiply the spin-orbit coupling term in the followings: $$e^{i\pi J_y}\langle d_{xy}\uparrow|\xi L\cdot\sigma|p_{z}\downarrow\rangle$$; where $$e^{i\pi J_y}$$ is the time reversal operator, this term should change the sign. Am I correct? If so, how to choose $J_y$, the y-component of the angular momentum operator, for different term? I think that this operator varies with different orbital $p_z$,$d_{xy}$. Thank you. $\endgroup$
    – Kieran
    Commented Apr 3, 2022 at 17:40

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