Symmetry of the Green-Keldysh-Nambu function

[Paul159]

Hello,

I would like to understand a relation of this article by Volkov (eq. 4).
Let's define the Green function $$ G^{ij}_{ab} (1,2) = -i \langle T_c \Psi_a (1_i) \Psi_b (2_j) \rangle $$ where $a,b = (1,2)$ are the spin indices and $i,j = (1,2)$ are the indices for the Keldysh contour ; $\Psi_1 = \Psi_\uparrow$, $\Psi_2 = \Psi_\downarrow^\dagger$. $T_c$ is the contour ordering.

Now I cite Volkov : "if the system contains no fields acting directly on the spins and if the spin-orbit interaction can be neglected we have :
$$ G^{ij*}_{ab} (1,2) = G^{\bar{ij}}_{\bar{ab}} (1,2) $$ where $\bar{1} = 2, \bar{2} = 1$. "

I don't understand why this last symmetry is true. For example I get with the last equation that
$$ \langle \Psi_\uparrow (1_1)^\dagger \Psi_\uparrow(2_1) \rangle = \langle \Psi_\downarrow (1_2)^\dagger \Psi_\downarrow(2_2) \rangle ,$$ where 1 is after 2 on the first branch.

Thanks in advance for any help.

 

[eljose79]

Thank you for your question about the relation in Volkov's article. It appears that the symmetry you are referring to is related to the time-reversal symmetry of the system. In the absence of external fields and spin-orbit interactions, the time-reversal symmetry of the system is preserved. This means that the Green function should be invariant under the transformation $\bar{1} = 2, \bar{2} = 1$, as stated in the equation you cited.

To understand this symmetry, let's consider the time evolution of the system. Since there are no external fields acting directly on the spins, the time evolution of the spin operators will be the same on both branches of the Keldysh contour. This means that the Green function for spin-up and spin-down particles should be the same on both branches, leading to the symmetry stated in Volkov's article.

To address your example, the equation you wrote is not quite correct. The correct equation should be $$ \langle \Psi_\uparrow (1_1)^\dagger \Psi_\uparrow(2_1) \rangle = \langle \Psi_\downarrow (1_2)^\dagger \Psi_\downarrow(2_1) \rangle ,$$ where the spin operators on the first branch are the same for both spin-up and spin-down particles. This is consistent with the time-reversal symmetry of the system.

I hope this explanation helps you understand the relation in Volkov's article. If you have any further questions, please don't hesitate to ask. Thank you for your interest in this topic.