When we can justify the response to be hold in terms of symmetry?

Question

In this paper, the authors argue the following response relation $$j_a=\sigma_{abc}E_bE_c \tag{1}$$ can occur in systems with $\mathcal{P}\mathcal{T}$-symmetry with $\mathcal{T}$ the time-reversal symmetry and $\mathcal{P}$ the inversion symmetry, where we assume that $\sigma$ is free of the relaxation time $\tau$. This is due to $$\mathcal{T}j_a \rightarrow -j_a, \mathcal{T}E_b \rightarrow E_b, \mathcal{P}j_a=-j_a, \mathcal{P}E_b\rightarrow -E_b, $$ therefore, we have $$\mathcal{P}\mathcal{T} \sigma_{abc}=\sigma_{abc}$$ which means the response relation can be appeared in $\mathcal{P}\mathcal{T}$-symmetric systems. The analysis seems reasonable. Following the same strategy, let's look at the response relation for the spin Hall effect: $$j^a_b=\sigma_{abc}^{SHE}E_c \tag{2},$$ in which the spin current $j_b^a$ and $E_c$ remain unchanged under $\mathcal{T}$, Eq.(2) can exist in $\mathcal{T}$-symmetric systems. Now let's consider the inversion symmetry $\mathcal{P}$, we find that $j_b^a$ and $E_c$ also remain unchanged under $\mathcal{P}$, and we conclude that Eq.(2) can exist in $\mathcal{P}$-symmetric systems.

Therefore, I conclude that the spin Hall effect can only exist in systems with both $\mathcal{P}$ and $\mathcal{T}$?

Is this right? I guess the answer is no.

Because in this paper, the authors predicted the spin Hall effect in the Weyl semimetals TaAs family, in which $\mathcal{T}$ is preserved but $\mathcal{P}$ is broken.

Following this, can I think that the spin hall effect can also be supported by the magnetic Weyl semimetals with $\mathcal{P}$-symmetry?

Answer 1

• The first part:

Eq.(1) manifest the conductivity $\sigma_{abc}$ is a $\mathcal{T}$-odd and $\mathcal{P}$-odd tensor, which indicates that Eq.(1) can not appear in systems with $\mathcal{T}$-symmetry or $\mathcal{P}$-symmetry. However, by Eq.(1), we find that $\sigma_{abc}$ is a $\mathcal{P}\mathcal{T}$-even tensor, which means this response relation is supported by systems

• without $\mathcal{T}$-symmetry and $\mathcal{P}$-symmetry but with the combined $\mathcal{P}\mathcal{T}$-symmetry.
• without $\mathcal{T}$-symmetry and $\mathcal{P}$-symmetry as well as the combined $\mathcal{P}\mathcal{T}$-symmetry.

Note that the odd and even means that $$\mathcal{P}\sigma_{abc}=-\sigma_{abc},\mathcal{T}\sigma_{abc}=-\sigma_{abc},\mathcal{P}\mathcal{T}\sigma_{abc}=\sigma_{abc},$$ which can be easily derived just from the Eq.(1).

• The second part:

It is easily shown that $\sigma^{SHE}_{abc}$ is a $\mathcal{T}$-even and $\mathcal{P}$-even tensor. Therefore, Eq.(2) is supported by systems with/without $\mathcal{P}$-symmetry ($\mathcal{T}$-symmetry).