.. _ug_tb_sh_4-2:

************
(4, 2) terms
************

Second type of quadlinear terms, in which three of the four spin operators are associated
with the same site, and the fourth spin operator is associated with a different site. In
this page we imply that
:math:`\boldsymbol{r}_{\mu,\alpha_1} \neq \boldsymbol{r}_{\mu + \nu_2, \alpha_2}`.

.. math::
    C_{4, 2}
    \sum_{\substack{\mu, \nu_2, \\ \alpha_1, \alpha_2, \\ i_1, i_2, i_3, i_4}}
    \Biggl(
        &
        J^{i_1, i_2, i_3, i_4}_{0, 0, \nu_2; \alpha_1, \alpha_1, \alpha_1, \alpha_2}
        S_{\mu, \alpha_1}^{i_1}
        S_{\mu, \alpha_1}^{i_2}
        S_{\mu, \alpha_1}^{i_3}
        S_{\mu + \nu_2, \alpha_2}^{i_4}
        +\\&+
        J^{i_1, i_2, i_3, i_4}_{0,\nu_2, 0; \alpha_1,\alpha_1,\alpha_2,\alpha_1}
        S_{\mu, \alpha_1}^{i_1}
        S_{\mu, \alpha_1}^{i_2}
        S_{\mu + \nu_2, \alpha_2}^{i_3}
        S_{\mu, \alpha_1}^{i_4}
        +\\&+
        J^{i_1, i_2, i_3, i_4}_{\nu_2, 0, 0; \alpha_1, \alpha_2, \alpha_1, \alpha_1}
        S_{\mu, \alpha_1}^{i_1}
        S_{\mu + \nu_2, \alpha_2}^{i_2}
        S_{\mu, \alpha_2}^{i_3}
        S_{\mu, \alpha_2}^{i_4}
        +\\&+
        J^{i_1, i_2, i_3, i_4}_{\nu_2, \nu_2, \nu_2; \alpha_1, \alpha_2, \alpha_2, \alpha_2}
        S_{\mu, \alpha_1}^{i_1}
        S_{\mu + \nu_2, \alpha_2}^{i_2}
        S_{\mu + \nu_2, \alpha_2}^{i_3}
        S_{\mu + \nu_2, \alpha_2}^{i_4}
    \Biggr)


Relevant API
============

*   :py:attr:`magnopy.Convention.c42`

    Convention constant.

*   :py:meth:`magnopy.SpinHamiltonian.add`

    Method to add the parameter to the Hamiltonian.

*   :py:meth:`magnopy.SpinHamiltonian.remove`

    Method to remove the parameter from the Hamiltonian.

*   :py:attr:`magnopy.SpinHamiltonian.p42`

    An iterator over the parameters already added to the Hamiltonian.