Equivalent parameters

An example

Let one consider the bilinear Hamiltonian that contains (among other) the following term

\[\mathcal{H} = ... + J_{\nu_2; \alpha_1, \alpha_2}^{x,y} S_{\mu, \alpha_1}^x S_{\mu+\nu_2, \alpha_2}^y + J_{-\nu_2; \alpha_2, \alpha_1}^{y,x} S_{\mu+\nu_2, \alpha_2}^y S_{\mu, \alpha_1}^x + ...\]

Now, in the case when \(\boldsymbol{r}_{\mu, \alpha_1} \ne \boldsymbol{r}_{\mu+\nu_2, \alpha_2}\), the spin operators \(S_{\mu, \alpha_1}^x\) and \(S_{\mu+\nu_2, \alpha_2}^y\) commute, thus the Hamiltonian can be rewritten as

\[\mathcal{H} = ... + \left( J_{\nu_2; \alpha_1, \alpha_2}^{x,y} + J_{-\nu_2; \alpha_2, \alpha_1}^{y,x} \right) S_{\mu, \alpha_1}^x S_{\mu+\nu_2, \alpha_2}^y + ...\]

You can see that the physics of the Hamiltonian is determined only by the sum \(J_{\nu_2; \alpha_1, \alpha_2}^{x,y} + J_{-\nu_2; \alpha_2, \alpha_1}^{y,x}\). Therefore, that sum can be distributed between the parameters arbitrarily. We call those two components of the parameters to be equivalent. More generally, a set of parameters is called to be a set of equivalent parameters (or an equivalent set) if the physics of the Hamiltonian is determined solely by the sum of the (components of the) parameters from that set.

In the case of the bilinear term, every equivalent set contains two parameters:

\[J_{\nu_2; \alpha_1, \alpha_2}^{i_1, i_2} \quad \text{and} \quad J_{-\nu_2; \alpha_2, \alpha_1}^{i_2, i_1}\]

In the case of the other types of terms, the equivalent sets can contain more that two parameters. All equivalent sets are listed in the supplementary notes of the paper about Magnopy (equations (S.21)—(S.27)).

Symmetrization

As one is free to distribute the sum of the parameters from the equivalent set arbitrarily, one can choose the parameters from the equivalent set to be equal to each other. For example, for the bilinear term, one can choose

\[J_{\nu_2; \alpha_1, \alpha_2}^{i_1, i_2} = J_{-\nu_2; \alpha_2, \alpha_1}^{i_2, i_1}\]

We call such choice a case of symmetrized parameters.

The user is free to input non-symmetrized parameters to the magnopy.SpinHamiltonian class. However, Magnopy will symmetrize them when it is necessary.

Note

The symmetrization of the parameters do not place any restrictions on the components of the vector/matrix/tensor of each individual parameter. It only relates components of the vector/matrix/tensor of different parameters. For instance, it does not forbid an antisymmetric Dzyaloshinskii-Moriya interaction.

Connection with multiple_counting

The choice of the distribution of the sum within the equivalent set is only relevant when more than one equivalent interaction is included in the summation over the magnetic sites. Therefore, the choice of the distribution of the sum within the equivalent set is only relevant when multiple_counting = True.