Closed ZongYongyue closed 1 month ago
Hi Yong-Yue Zong,
Thanks for the suggestion! I am definitely very happy to add this, would you have a suggestion for a good name? In hindsight, naming these functions with a single letter is probably not very future-proof, as S
could be many different things...
Best, Lukas
How about Sₑ
$\to\hat{S}$ , Sₑ²
$\to\hat{S}^2$ and Sₑ_exchange
$\to\hat{S}_i \cdot \hat{S}_j$? Where e
has considered the style in MPSKitModels
that indicates a spin 1/2 fermionic system and Sₑ
can be used to distinguish from the spin operators such as Sˣ, S⁺, S_exchange
in spin systems.
I would also need a non-unicode version, as there still are some systems (terminals, editors, ...) where these are very hard to type.
I don't have too much time the next week, any chance you could file a PR with this?
Of course
Hi Lukas,
Thanks for your package! I suggest to add a spin irreducible operator $\hat{S}=\sum{\sigma\sigma'}f^\dagger\sigma\vec{\sigma}f_{\sigma'}=(-S^{+}/\sqrt{2},S^{z},S^{-}/\sqrt{2})$ in a spinful fermionic system with U(1) charge and SU(2) spin symmetry. This could be beneficial as we consider magnetic properties in electronic systems. With Wigner–Eckart theorem, we have
$$\langle 0; 1/2, -1/2| S^{-} /\sqrt{2}|0;1/2,1/2\rangle = C^{1/2}_{1/2,1}[2,1,1]\langle 0;1/2|S^{0;1}|0;1/2\rangle $$
where the quantum number are labeled as $(q_n; S, Sz)$ and $C^{1/2}{1/2,1}$ is the CG coefficients with dimension $=2\times 3 \times 2.$ The other two equations about $S^{z}$ and $S^{+}$ are equivalent with this one because the reduced matrix element $\langle 0;1/2|S^{0;1}|0;1/2\rangle$ is independent of quantum number $S_z$. Here, we have $\langle 0,1/2|S^{0;1}|0;1/2\rangle = \sqrt{3}/2$ .
We can construct the spin irreducible operator $\hat{S}$ as
Thank you again for your nice package!
Sincerely yours Yong-Yue Zong