Quanty - documentation:standard_operators:spin
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2026-04-30T14:29:14+00:00Quanty
https://www.quanty.org/
https://www.quanty.org/_media/wiki/dokuwiki.svgtext/html2025-11-20T02:29:34+00:00Anonymous (anonymous@undisclosed.example.com)Smin
https://www.quanty.org/documentation/standard_operators/spin/smin?rev=1763605774&do=diff
Smin
The $S^-$ operator is defined as:
\begin{equation}
S^- = \sum_{\tau} a^{\dagger}_{\tau\downarrow}a^{\phantom{\dagger}}_{\tau\uparrow}.
\end{equation}
The equivalent operator in Quanty is created by:
OppSmin = NewOperator("Smin", NF, IndexUp, IndexDn)
Table of contents
spin indextext/html2025-11-20T02:29:34+00:00Anonymous (anonymous@undisclosed.example.com)Splus
https://www.quanty.org/documentation/standard_operators/spin/splus?rev=1763605774&do=diff
Splus
The $S^+$ operator is defined as:
\begin{equation}
S^+ = \sum_{\tau} a^{\dagger}_{\tau\uparrow}a^{\phantom{\dagger}}_{\tau\downarrow}.
\end{equation}
The equivalent operator in Quanty is created by:
OppSplus = NewOperator("Splus", NF, IndexUp, IndexDn)
Table of contents
spin indextext/html2025-11-20T02:29:34+00:00Anonymous (anonymous@undisclosed.example.com)Ssqr
https://www.quanty.org/documentation/standard_operators/spin/ssqr?rev=1763605774&do=diff
Ssqr
The $S^2$ operator is defined as $S_x^2+S_y^2+S_z^2$ or:
\begin{eqnarray}
S^2 &=& \sum_{\tau,\sigma} \frac{3}{4}a^{\dagger}_{\tau,\sigma}a^{\phantom{\dagger}}_{\tau,\sigma} \\
\nonumber &+& \sum_{\tau_1,\tau_2}\sum_{\sigma_1,\sigma_2} \, -\sigma_1\sigma_2 \, a^{\dagger}_{\tau_1\sigma_1}a^{\dagger}_{\tau_2\sigma_2}a^{\phantom{\dagger}}_{\tau_1\sigma_1}a^{\phantom{\dagger}}_{\tau_2\sigma_2} \\
\nonumber &&\quad\quad\quad\quad - a^{\dagger}_{\tau_1\uparrow}a^{\dagger}_{\tau_2\downarrow}a^{\ph…text/html2025-11-20T02:29:34+00:00Anonymous (anonymous@undisclosed.example.com)Spin operators (S)
https://www.quanty.org/documentation/standard_operators/spin/start?rev=1763605774&do=diff
Spin operators (S)
The spin operators need as a basis definition two lists of equal length. The $\textit{up}$ list indexes the orbitals with spin up, the $\textit{down}$ list indexes the orbitals with spin down. I will generally use the even numbers for spin down and the odd numbers for spin up. (note that this yields the same basis definition as used in the $\textit{Mathematica}$spin indextext/html2025-11-20T02:29:35+00:00Anonymous (anonymous@undisclosed.example.com)Sx
https://www.quanty.org/documentation/standard_operators/spin/sx?rev=1763605775&do=diff
Sx
The $S_x$ operator is defined as:
\begin{equation}
S_x = \sum_{\tau} \frac{1}{2}\left(a^{\dagger}_{\tau\uparrow}a^{\phantom{\dagger}}_{\tau\downarrow} + a^{\dagger}_{\tau\downarrow}a^{\phantom{\dagger}}_{\tau\uparrow}\right).
\end{equation}
The equivalent operator in Quanty is created by:
OppSx = NewOperator("Sx", NF, IndexUp, IndexDn)
Table of contents
spin indextext/html2025-11-20T02:29:34+00:00Anonymous (anonymous@undisclosed.example.com)Sy
https://www.quanty.org/documentation/standard_operators/spin/sy?rev=1763605774&do=diff
Sy
The $S_y$ operator is defined as:
\begin{equation}
S_y = \sum_{\tau} \frac{\imath}{2}\left(-a^{\dagger}_{\tau\uparrow}a^{\phantom{\dagger}}_{\tau\downarrow} + a^{\dagger}_{\tau\downarrow}a^{\phantom{\dagger}}_{\tau\uparrow}\right).
\end{equation}
The equivalent operator in Quanty is created by:
OppSy = NewOperator("Sy", NF, IndexUp, IndexDn)
Table of contents
spin indextext/html2025-11-20T02:29:35+00:00Anonymous (anonymous@undisclosed.example.com)Sz
https://www.quanty.org/documentation/standard_operators/spin/sz?rev=1763605775&do=diff
Sz
The $S_z$ operator is defined as:
\begin{equation}
S_z = \sum_{\tau} \frac{1}{2}\left(a^{\dagger}_{\tau\uparrow}a^{\phantom{\dagger}}_{\tau\uparrow} - a^{\dagger}_{\tau\downarrow}a^{\phantom{\dagger}}_{\tau\downarrow}\right).
\end{equation}
The equivalent operator in Quanty is created by:
OppSz = NewOperator("Sz",NF,IndexUp,IndexDn)
Table of contents
spin index