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I am studying the rashba spin orbit interaction in graphene

I know the effective Hamiltonian for RSOI in the presence of a perpendicular electric field ($E_z$) is $$ H_{RSO}=\frac{\alpha}{\hbar}(P_x\sigma_y-P_y\sigma_x) \tag{1} $$ where P is the momentum vector and σ is the pauli matrix vector but there is also another form of effective Hamiltonian $$ H_{RSO}=\frac{\lambda R}{2}(\tau\sigma_xS_y−\sigma_yS_x) \tag{2} $$ where $\tau=1$ and $-1$ in K and K' point. $\sigma_x$ and $\sigma_y$ are pseudospin and $S_x$ and $S_y$ are spin angular momentum in spin space

my question is that are Eq. (1) and Eq. (2) equivalent? and how eq.(2) is derived?

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    $\begingroup$ you need to supply more details. For example - I'm guessing that you are talking specifically about graphene? what are the $\sigma_i$ and $S_i$ etc.? $\endgroup$
    – user275556
    Commented Nov 9, 2021 at 9:39
  • $\begingroup$ yes i am studying rsoi in graphene. I edited the question and added more details thank you $\endgroup$
    – fa sh
    Commented Nov 9, 2021 at 9:59
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    $\begingroup$ Please check the meaning of $\sigma_{x,y}$ and $S_{x,y}$ in Eq. (2) - one should be spin operator and the other orbital momentm (angular or linear)... otherwise it wouldn't be spin-orbit coupling. $\endgroup$
    – Roger V.
    Commented Nov 9, 2021 at 10:05

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