Theory notes#
In this page we write some formulas that did not make it neither to the publication about magnopy nor to the user guide, but should be listed somewhere for the convenience of the developers.
Classical energy#
The formulas for the classical energy are written for the case of non-normalized spins, all other convention properties being arbitrary.
\[\begin{split}E^{(0)}
=&
\,C_1
\sum_{\alpha, i}
J_1^i(\boldsymbol{r}_{\alpha})
z^i_{\alpha}
S_{\alpha}
+\\&+
C_{2,1}
\sum_{\alpha, i,j}
J_{2,1}^{ij}(\boldsymbol{r}_{\alpha})
z^i_{\alpha}
z^j_{\alpha}
(S_{\alpha})^2
+\\&+
C_{2,2}
\sum_{\substack{\alpha, \beta, \nu, \\ i,j}}
J_{2,2}^{ij}(\boldsymbol{r}_{\nu,\alpha\beta})
z^i_{\alpha}
z^j_{\beta}
S_{\alpha}
S_{\beta}
+\\&+
C_{3, 1}
\sum_{\substack{\alpha, \\ i, j, u}}
J^{iju}_{3, 1}(\boldsymbol{r}_{\alpha})
z^i_{\alpha}
z^j_{\alpha}
z^u_{\alpha}
(S_{\alpha})^3
+\\&+
C_{3, 2}
\sum_{\substack{\alpha, \beta, \nu, \\ i, j, u}}
J^{iju}_{3, 2}(\boldsymbol{r}_{\nu,\alpha\beta})
z^i_{\alpha}
z^j_{\alpha}
z^u_{\beta}
(S_{\alpha})^2
S_{\beta}
+\\&+
C_{3, 3}
\sum_{\substack{\alpha, \beta, \gamma, \\ \nu, \lambda, \\ i, j, u}}
J^{iju}_{3, 3}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})
z^i_{\alpha}
z^j_{\beta}
z^u_{\gamma}
S_{\alpha}
S_{\beta}
S_{\gamma}
+\\&+
C_{4, 1}
\sum_{\substack{\alpha, \\ i, j, u, v}}
J_{4, 1}^{ijuv}(\boldsymbol{r}_{\alpha})
z^i_{\alpha}
z^j_{\alpha}
z^u_{\alpha}
z^v_{\alpha}
(S_{\alpha})^4
+\\&+
C_{4, 2, 1}
\sum_{\substack{\alpha, \beta, \nu, \\ i, j, u, v}}
J_{4, 2, 1}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta})
z^i_{\alpha}
z^j_{\alpha}
z^u_{\alpha}
z^v_{\beta}
(S_{\alpha})^3
S_{\beta}
+\\&+
C_{4, 2, 2}
\sum_{\substack{\alpha, \beta, \nu, \\ i, j, u, v}}
J_{4, 2, 2}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta})
z^i_{\alpha}
z^j_{\alpha}
z^u_{\beta}
z^v_{\beta}
(S_{\alpha})^2
(S_{\beta})^2
+\\&+
C_{4, 3}
\sum_{\substack{\alpha, \beta, \gamma, \\ \nu, \lambda, \\ i, j, u, v}}
J_{4, 3}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})
z^i_{\alpha}
z^j_{\alpha}
z^u_{\beta}
z^v_{\gamma}
(S_{\alpha})^2
S_{\beta}
S_{\gamma}
+\\&+
C_{4, 4}
\sum_{\substack{\alpha, \beta, \gamma, \varepsilon, \nu, \lambda, \rho, \\ \\ i, j, u, v}}
J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})
z^i_{\alpha}
z^j_{\beta}
z^u_{\gamma}
z^v_{\varepsilon}
S_{\alpha}
S_{\beta}
S_{\gamma}
S_{\varepsilon}\end{split}\]
Renormalized parameters#
\[\begin{split}J^{\prime, i}(\boldsymbol{r}_{\alpha})
=&
C_1
J^i_1(\boldsymbol{r}_{\alpha})
+\\&+
2C_{2,1}
\sum_{j}
J^{ij}_{2,1}(\boldsymbol{r}_{\alpha})
z^j_{\alpha}S_{\alpha}
+\\&+
2C_{2,2}
\sum_{\beta, \nu, j}
J^{ij}_{2,2}(\boldsymbol{r}_{\nu,\alpha\beta})
z^j_{\beta}S_{\beta}
+\\&+
3C_{3, 1}
\sum_{j, u}
J^{iju}_{3, 1}(\boldsymbol{r}_{\alpha})
z^j_{\alpha}
z^u_{\alpha}
S_{\alpha}
S_{\alpha}
+\\&+
3C_{3, 2}
\sum_{\substack{\beta, \nu, \\ j, u}}
J^{iju}_{3, 2}(\boldsymbol{r}_{\nu,\alpha\beta})
z^j_{\alpha}
z^u_{\beta}
S_{\alpha}
S_{\beta}
+\\&+
3C_{3, 3}
\sum_{\substack{\beta, \gamma, \\ \nu, \lambda, \\ j, u}}
J^{iju}_{3, 3}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})
z^j_{\beta}
z^u_{\gamma}
S_{\beta}
S_{\gamma}
+\\&+
4C_{4, 1}
\sum_{\substack{j, u, v}}
J_{4, 1}^{ijuv}(\boldsymbol{r}_{\alpha})
z^j_{\alpha}
z^u_{\alpha}
z^v_{\alpha}
S_{\alpha}
S_{\alpha}
S_{\alpha}
+\\&+
4C_{4, 2, 1}
\sum_{\substack{\beta, \nu, \\ j, u, v}}
J_{4, 2, 1}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta})
z^j_{\alpha}
z^u_{\alpha}
z^v_{\beta}
S_{\alpha}
S_{\alpha}
S_{\beta}
+\\&+
4C_{4, 2, 2}
\sum_{\substack{\beta, \nu, \\ j, u, v}}
J_{4, 2, 2}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta})
z^j_{\alpha}
z^u_{\beta}
z^v_{\beta}
S_{\alpha}
S_{\beta}
S_{\beta}
+\\&+
4C_{4, 3}
\sum_{\substack{\beta, \gamma \\ \nu, \lambda, \\ j, u, v}}
J_{4, 3}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})
z^j_{\alpha}
z^u_{\beta}
z^v_{\gamma}
S_{\alpha}
S_{\beta}
S_{\gamma}
+\\&+
4C_{4, 4}
\sum_{\substack{\beta, \gamma, \varepsilon, \\ \nu, \lambda, \rho, \\ j, u, v}}
J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})
z^j_{\beta}
z^u_{\gamma}
z^v_{\varepsilon}
S_{\beta}
S_{\gamma}
S_{\varepsilon}\end{split}\]
\[\begin{split}J^{\prime, ij}(\boldsymbol{r}_{\nu,\alpha\beta})
=&
C_{2, 2}
J^{ij}_{2,2}(\boldsymbol{r}_{\nu,\alpha\beta})+\\&+
\delta_{\alpha,\beta}
\Biggl(
2C_{2,1}
J^{ij}_{2,1}(\boldsymbol{r}_{\alpha})
+\\&\phantom{+\delta_{\alpha,\beta}\Biggl(}+
3C_{3, 1}
\sum_{u}
J^{iju}_{3, 1}(\boldsymbol{r}_{\alpha})
z^u_{\alpha}
S_{\alpha}
+\\&\phantom{+\delta_{\alpha,\beta}\Biggl(}+
6C_{4, 1}
\sum_{u, v}
J_{4, 1}^{ijuv}(\boldsymbol{r}_{\alpha})
z^u_{\alpha}
z^v_{\alpha}
S_{\alpha}
S_{\alpha}
\Biggr)
+\\&+
3C_{3, 2}
\sum_{\nu, u}
J^{iuj}_{3, 2}(\boldsymbol{r}_{\nu,\alpha\beta})
z^u_{\alpha}
S_{\alpha}
+\\&+
3C_{3, 3}
\sum_{\gamma, \lambda, u}
J^{iju}_{3, 3}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})
z^u_{\gamma}
S_{\gamma}
+\\&+
6C_{4, 2, 1}
\sum_{u, v}
J_{4, 2, 1}^{iuvj}(\boldsymbol{r}_{\nu,\alpha\beta})
z^u_{\alpha}
z^v_{\alpha}
S_{\alpha}
S_{\alpha}
+\\&+
6C_{4, 2, 2}
\sum_{u, v}
J_{4, 2, 2}^{iujv}(\boldsymbol{r}_{\nu,\alpha\beta})
z^u_{\alpha}
z^v_{\beta}
S_{\alpha}
S_{\beta}
+\\&+
6C_{4, 3}
\sum_{\substack{\gamma, \lambda, \\ u, v}}
J_{4, 3}^{iujv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma})
z^u_{\alpha}
z^v_{\gamma}
S_{\alpha}
S_{\gamma}
+\\&+
6C_{4, 4}
\sum_{\substack{\gamma, \varepsilon, \\ \lambda, \rho, \\ u, v}}
J_{4, 4}^{ijuv}(\boldsymbol{r}_{\nu,\alpha\beta}, \boldsymbol{r}_{\lambda,\alpha\gamma}, \boldsymbol{r}_{\rho,\alpha\varepsilon})
z^u_{\gamma}
z^v_{\varepsilon}
S_{\gamma}
S_{\varepsilon}\end{split}\]