# Why linear response is absent in a non-centrosymmetric system with time reversal symmetry?

In this paper, it is mentioned that a non-centrosymmetric system with time-reversal symmetry doesn't have a linear response. It is actually a consequence of the Onsager reciprocal theorem.

But I didn't understand the theorem properly. Can anyone give an intuitive picture of why the linear response is absent in a non-centrosymmetric system with time-reversal symmetry?

• Could you please check Box-1 Dec 3 '20 at 16:05
• I think I am asking why the linear response is forbidden in a non-centrosymmetric system with $T$ ? Dec 3 '20 at 16:10
• I see. Your post may be clearer if you add "diagonal" as that is what it points to. Dec 3 '20 at 16:14
• I am sorry. I didn't get you. Where should I have to add "diagonal"? If you don't mind could you please do the edits for me? Dec 3 '20 at 16:18
• It is possible for the system with $\hat T$ to have a linear off-diagonal response (circular dichroism) but not a linear diagonal response. As for the formulation of the theorem, the identity $\sigma_{\alpha\beta}(q,w,B)=\sigma_{\beta\alpha}(-q,w,-B)$ follows as the simultaneous reversal "changes the $q$-representation of the Hamiltonian and the wave function to their complex conjugates" (Kubo, 1957). The algebraic proof is a simple comparison of real and imaginary parts of the function. Dec 3 '20 at 16:31

As to why nonreciprocal linear response is forbidden (at least, as mentioned in the paper, for the diagonal elements). This comes from these Onsager reciprocal relations, which relates some linear response function to a similar, time reversed response function: $$\tag{1}K_{AB}(q)=\epsilon_A\epsilon_BK_{BA}(-q)$$
This just says that the response function for operators A and B (A input, B output) is equal to the response function for B and A (B input, A output) with the wavevector flipped and the sign flipped based on the time symmetry of A and B. For a diagonal element (say xx), reversing the order of A and B doesn't matter (it's still the xx element of the same tensor) and assuming time reversal symmetry for each operator, this reduces to $$\tag{2}K_{A_xB_x}(q)-K_{A_xB_x}(-q)=0$$ This is exactly the nonreciprocal response for this element, so we can see that it will always vanish for a diagonal element. For an off-diagonal (say xy), reversing the order of A and B does matter (the reciprocal theorem relates the xy and time reversed yx elements of the tensor) $$\tag{3}K_{A_xB_y}(q)-K_{B_yA_x}(-q)=0$$ For the nonreciprocal response to be zero, the second term would be $$K_{B_xA_y}(-q)$$. The relation doesn't say anything about the nonreciprocal linear response in this case, so in general these elements will be nonzero.