Attempt at generalization (#4)

(Crystal) structure

A Hamiltonian is defined on a real-space structure (i. e. crystal, lattice, molecule, ion, etc.). A natural first step is to formally describe such structure.

First, imagine an arbitrary periodic lattice in d-dimensions, defined by the \(d\) lattice vectors \(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_d\). Label the cells of such lattice with the subscript indices \(\mu\) (or \(\mu_1\), \(\mu_2\), and so on if more than one cell index is required). In such way, the sum of the form

\[\sum_{\mu_1, \mu_2} \ldots\]

indicates that its argument is summed over all cells of the lattice twice. In other words, index \(\mu_1\) runs over all cells of the lattice and for each value of \(\mu_1\), index \(\mu_2\) also runs over all cells of the lattice.

Second, assume that there are \(N\) (magnetic) sites in each cell of the lattice. Label each site in the lattice with an index \(\alpha\) (or \(\alpha_1\), \(\alpha_2\), and so on if more than one site index is required).

With the definitions from above, the position of an arbitrary site in such structure is defined by the radius vector

\[\mathbf{r}_{\mu; \alpha} = \mathbf{r}_{\mu} + \mathbf{r}_{\alpha}\]

Finally, the cartesian components are denoted with the superscript indices \(i\) (or \(i_1\), \(i_2\), and so on if more than one Cartesian component index is required).

\[r_{\mu; \alpha}^{i} = r_{\mu}^{i} + r_{\alpha}^{i}\]

for \(i = 1, \ldots, d\) (or \(i = x, y, z\) if \(d = 3\)).

In the following one shall keep in mind the usual 3D space (\(d = 3\)).

(Magnetic) sites

Once the structure is defined, imagine that each site has an "entity" associated with it. Label such an entity with the capital letter \(X\). The interpretation of the letter \(X\) depends on the term of the Hamiltonian, which it enters. For example, it can be a spin vector, a spin operator, an angular momentum operator, a Stevens operator, a Wybourne operator, an external magnetic field and so on.

Common for all interpretations of the entity \(X\) are the following properties

  • Each entity have \(n\) components.

    \[X^{i}\]

    where \(i\) runs from \(1\) to \(n\). In the simplest case those are Cartesian components (\(i = x, y, z\)). However, in some cases those can have different meaning.

  • Each entity has a well defined position in the real space.

    \[X_{\mu; \alpha}\]

    As each entity is associated with a site, its position is simply the position of tha site.

"Extra" variables

Some terms of the Hamiltonian contain special variables, that may not depend on the site indices. For example, strain tensor in McPhase or an external effect, as a magnetic flux density. Such variables

Those terms include variables, that are expected to be described as entities, but are either associated with a crystal as a whole or are uniform for all sites. Therefore, the association with a particular site is not natural for such variables. To account for such variables introduce a special value for the indices \((\mu, \alpha)\), for example \((\mu, \alpha) \in \{\_1, \_2, \_3, \dots\}\).

Note

The form of the values — "_<integer>" — is a suggestion. The actual label can be decided during the design stage for the details of the format.

Then, such variables can be formally written as entities as well. The indication of how those terms shall be interpreted shall be designed as a detail of the format. The importance for the generalized algebraic form is the possibility of having a special index value.

Note

Consider a Zeeman term (let \(g\mu_B = 1\)) \(\sum_{\mu, \alpha, i} B^{i} S_{\mu; \alpha}^{i}\). A spin operator/vector is an entity, that have a clear association with a site \(X^{i}_{\mu; \alpha} \equiv S^i_{\mu; \alpha}\). However, the magnetic flux density \(B^{i}\) is typically uniform for all sites and therefore does not have a clear association with a particular site. Using a concept of special values for the site indices, one can write \(X^i_{\_1} \equiv B^i\). Then the Zeeman term can be written as

\[\sum_{\mu_1, \alpha_1, \mu_2, \alpha_2, i} X^i_{\mu_1, \alpha_1} \cdot X^{i}_{\mu_2; \alpha_2}\]

In such a sum one understands that the file would include only blocks with \((\mu_1, \alpha_1) = \_1\), but the algebraic form is written in such a way that \(X^i_{\_1}\) and \(X^{i}_{\mu; \alpha}\) are treated on equal footing.

In the following, the meaning of the word "sites" is extended with such special value of its indices.

Interaction parameters

Then, an interaction between sites is described by the interaction parameters. Label such an interaction parameter with the capital letter \(V\). Each interaction parameter connects one or more entities. As with entities, the interpretation of the parameter \(V\) depends on the term of the Hamiltonian, which it enters. However, there are two common properties

  • Each interaction parameter has as many independent dimensions as the number of components of the entities that it connects. For example, an interaction parameter

    \[V^{i_1, i_2, \ldots, i_k}\]

    connects \(k\) entities.

  • Each interaction parameter depends on as many pairs of position indices and/or labels as the number of entities it connects.

    \[V_{\mu_1, \mu_2, \ldots, \mu_m; \alpha_1, \alpha_2, \ldots, \alpha_k}\]

Terms of the Hamiltonian

The Hamiltonian itself is constructed with the following properties in mind

  • Terms of the Hamiltonian can involve an arbitrary amount of entities.

  • Each combination of components of the entities can have a unique interaction parameter associated with it.

Consider an arbitrary term of the Hamiltonian. Let it involve \(k\) sites (distinct or not)

\[(\mu_1; \alpha_1), (\mu_2; \alpha_2), \ldots, (\mu_m; \alpha_m)\]

Each site has an entity associated with it

\[X_{\mu_1; \alpha_1}^{i_1}, X_{\mu_2; \alpha_2}^{i_2}, \ldots, X_{\mu_k; \alpha_k}^{i_k}\]

Then, the interaction parameter that connects those entities is labeled as

\[V_{\mu_1, \ldots, \mu_k; \alpha_1, \ldots, \alpha_k}^{i_1, \ldots, i_k}\]

Then the term of the Hamiltonian that involves those entities is written as

\[\begin{split}\mathcal{H}_k = C_k \sum_{\substack{\mu_1, \ldots, \mu_k, \\ \alpha_1, \ldots, \alpha_k, \\ i_1, \ldots, i_k}} V_{\mu_1, \ldots, \mu_k; \alpha_1, \ldots, \alpha_k}^{i_1, \ldots, i_k} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_2; \alpha_2}^{i_2} \cdot \ldots \cdot X_{\mu_k; \alpha_k}^{i_k}\end{split}\]

Hint

The constants of the form \(C_{k}\) are introduced to account for different conventions (see issue #2).

The Hamiltonian can include multiple terms of the same form, with distinct physical origin.

The Hamiltonian

Finally, everything is ready to write the Hamiltonian, which simply a sum over all possible terms of the form from above

\[\begin{split}\mathcal{H} = \sum_{k=1}^{\infty} \mathcal{H}_k = \sum_{k=1}^{\infty} C_k \sum_{\substack{\mu_1, \ldots, \mu_k, \\ \alpha_1, \ldots, \alpha_k, \\ i_1, \ldots, i_k}} V_{\mu_1, \ldots, \mu_k; \alpha_1, \ldots, \alpha_k}^{i_1, \ldots, i_k} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_2; \alpha_2}^{i_2} \cdot \ldots \cdot X_{\mu_k; \alpha_k}^{i_k}\end{split}\]

Mapping to the general form

Now, consider the terms of the Hamiltonian that are present in the "zoo" and map each one of them to the general form above.

The pages below discuss the mapping of each term in detail.