Elastic energy¶
Elastic energy term involves two entities. Therefore, it can be expressed as a \(\mathcal{H}_2\) term.
\[\begin{split}\mathcal{H}_2
=
C_2
\sum_{\substack{\mu_1, \mu_2,\\ \alpha_1, \alpha_2,\\ i_1, i_2}}
V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}
X_{\mu_1; \alpha_1}^{i_1}
X_{\mu_2; \alpha_2}^{i_2}\end{split}\]
The summary of the mapping for each code is given in the table below.
Code |
|
|---|---|
\(\mathcal{H}\) |
\[\hat{H}
=
\dfrac{1}{2}
\sum_{\alpha\gamma=1-6}
c^{\alpha\gamma}
\epsilon_{\alpha}
\epsilon_{\gamma}\]
|
Renaming of indices |
\(\alpha \rightarrow i_1\), \(\gamma \rightarrow i_2\) |
\(C_2\) |
\(\dfrac{1}{2}\) |
\(V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1,i_2}\) |
\(\dfrac{c^{i_1 i_2}}{N^2 N_C^2}\) |
\(X_{\mu_1, \alpha_1}^{i_1}\) |
\(\epsilon_{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(\epsilon_{i_2}\) |
Note
\(i_1, i_2 = 1, \dots, 6\).
\(N\) is the number of sites in each cell and \(N_C\) is the number of cells in the system.