Time-reversal invariance and hermiticity of the pseudopotential

The pseudopotential studied in this work is a contact interaction built with derivative and spin operators. Furthermore, the choice concerning the formalism is the use of the spherical tensors. Under these assumptions, the general structure of the pseudopotential is based on the following building blocks,

$$\hat{V}_0 = \left[ \left[K'_{\tilde{n}'\tilde{L}'} K_{\tilde... ...]_{0} \hat{\delta}_{12}(\bm{r}'_1\bm{r}'_2;\bm{r}_1\bm{r}_2) .$$ (55)

The final coupling to a scalar ensures that $\hat{V}_0$ is invariant under space rotation Moreover, provided that $\tilde{n}'+\tilde{n}$ is even, it is also invariant under space-inversion. Now we proceed to explore another fundamental symmetry, the time-reversal, and later we also require the hermiticity of the pseudopotential.

The time-reversal operator $\hat{T}=-i\sigma_y\hat{K}$, where $\hat{K}$ is the complex conjugation in space representation, can be explicitly applied to the spherical-tensor representations of momentum and spin operators, Eqs. (3), (5), and (6), which gives the generic result for spherical tensors,

$$\hat{T}A_{\lambda\mu}\hat{T}^\dagger = T_A(-1)^{\lambda-\mu}A_{\lambda,-\mu},$$ (56)

where $T_A$ are numerical phase factors. In our case, we obtain $T_k=-1$ for the momentum operator and $T_{\sigma_v}=(-1)^v$ for the scalar ($v=0$) and vector ($v=1$) spin operators. Moreover, since the Clebsh-Gordan coefficients are real, rule (56) propagates through the angular momentum coupling, that is, if phase factors $T_A$ and $T_{A'}$ characterize tensors $A_\lambda$ and $A'_{\lambda'}$, respectively, then the coupled tensor,

$$A''_{\lambda''\mu''} = [A_\lambda A'_{\lambda'}]_{\lambda''\... ...''}_{\lambda\mu\lambda'\mu'} A_{\lambda\mu}A'_{\lambda'\mu'},$$ (57)

is characterized by the product of phase factors $T_{A''}= T_A T_{A'}$ (cf. Appendix B in Ref. [2]). Therefore, the coupled operators appearing in $\hat{V}_0$ (55) are characterized by the following values of phase factors,

$$T_{K'_{\tilde{n}'\tilde{L}'}} = (-1)^{\tilde{n}'} ,$$ (58)

$$T_{K_ {\tilde{n }\tilde{L} }} = (-1)^{\tilde{n} } ,$$ (59)

$$T_{S_ {v_{12}S}} = (-1)^{v_{12} } .$$ (60)

Finally, because the Dirac delta is real, for $\hat{V}_0$ we have,

$$T_{\hat{V}_0}=(-1)^{\tilde{n}'+\tilde{n }+v_{12}} ,$$ (61)

and by taking into account the space-inversion invariance, it boils down to

$$T_{\hat{V}_0}=(-1)^{v_{12}} .$$ (62)

This justifies the phase factor $i^{v_{12}}$ in the definition of the pseudopotential in Eq. (2), which ensures that for real parameters, all terms of the pseudopotential are time-even.

Now we can proceed to calculate the adjoint of the operator $\hat{V}_0$ (55) multiplied by the phase factor derived above, that is,

$$\left(i^{v_{12}}\hat{V}_0\right)^\dagger = (-i)^{v_{12}} \le... ...t]_{0} \hat{\delta}_{12}(\bm{r}'_1\bm{r}'_2;\bm{r}_1\bm{r}_2),$$ (63)

where we treat the space derivatives of the Dirac delta like ordinary numbers and the space variables had to be exchanged, $\bm{r}'_1\leftrightarrow\bm{r}_1$ and $\bm{r}'_2\leftrightarrow\bm{r}_2$.

Properties of generic spherical tensors under the complex and Hermitian conjugations are given by the following rules,

$$A_{\lambda\mu}^* = P_A(-1)^{\lambda-\mu}A_{\lambda,-\mu},$$ (64)

$$A_{\lambda\mu}^\dagger = H_A(-1)^{\lambda-\mu}A_{\lambda,-\mu},$$ (65)

where the phase factors $P_A$ and $H_A$ can be directly derived from definitions (3), (5), and (6), that is, $P_k=-1$ and $H_{\sigma_v}=+1$. These rules also propagate through the angular momentum coupling, that is, $P_{A''}= P_A P_{A'}$ and, for commuting operators, which is the case here, $H_{A''}= H_A H_{A'}$. Therefore, we have,

$$P_{\left[K_{\tilde{n}'\tilde{L}'} K'_{\tilde{n}\tilde{L}}\right]_{S}} =(-1)^{\tilde{n}'+\tilde{n}}=+1 ,$$ (66)

and

$$H_{\hat{S}_{v_{12} S}}=+1 .$$ (67)

Finally, the adjoint operator of Eq. (63) is given by

$$\left(i^{v_{12}}\hat{V}_0\right)^\dagger = (-i)^{v_{12}} \le... ...t]_{0} \hat{\delta}_{12}(\bm{r}'_1\bm{r}'_2;\bm{r}_1\bm{r}_2),$$ (68)

where the last equality results from flipping the order of coupling of the operators $K_{\tilde{n}'\tilde{L}'}$ and $K'_{\tilde{n }\tilde{L}}$, which brings out the phase factor of $(-1)^{S-\tilde{L}'-\tilde{L}}=(-1)^{S}$. Therefore, the time-even tensor $i^{v_{12}}\hat{V}_0$ is not self-adjoint, but we can hermitize it by using the expression given in Eq. (2).