Attempt at generalization (#5)

(Crystal) structure

A Hamiltonian is defined in the real space on a (periodic) structure, for example 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). Then, the sum of the form

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

implies the summation of its argument 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 as well.

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

Then, the position of an arbitrary site in such structure is given 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 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 \(d = 3\)).

Note

Component indices can denote generic components of the vector. In this section a special case of the Cartesian components is discussed.

In the next sections the component indices are not limited to Cartesian components.

(Magnetic) sites

Once the structure is defined, one is ready to discuss the variables that act on the system or operators/parameters of the system. For the sake of the generality call such object an "entity" and label it with the capital letter \(X\). An interpretation of such entity depends on the context of the particular term of the Hamiltonian that it enters. For example, it can denote a spin vector, a spin operator, an angular momentum operator, a Stevens operator, a Wybourne operator, an external magnetic field and so on.

Among all possible interpretations there are a few common properties of such entities.

  • Each entity has \(n\) components, which is denoted as

    \[X^{i}\]

    where \(i\) runs from \(1\) to \(n\). In the simplest case those are Cartesian components (\(i = x, y, z\)). However, the component indices can have other meaning as well.

  • Each entity is associated with exactly one site of the structure, which is denoted as

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

Hint

One site can have multiple entities associated with it. Think about an on-site anisotropy as an example.

Interaction parameters

After the structure is defined and populated with individual entities, the next step is to think about interaction between such entities.

Assume that an interaction parameter connects one or more entities and label it with the capital letter \(V\). As with entities, the interpretation of the parameter \(V\) depends on the term of the Hamiltonian, which it enters. However, there are a few common properties

  • Each interaction parameter has as many independent dimensions as the number of the entities that it connects.

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

For instance, an interaction parameter

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

connects \(k\) entities \(X^{i_1}_{\mu_1; \alpha_1}\), ..., \(X^{i_k}_{\mu_k; \alpha_k}\).

Hint

Size of each index \(i_1\), ..., \(i_k\) can be different.

Terms of the Hamiltonian

Before arriving at the general form of the Hamiltonian, 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

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

and the interaction parameter 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

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

Note

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

  • Any combination of entity's components can have an scalar individual parameter associated with it. Those scalar parameters are collected into an tensor of the interaction parameter \(V_{\mu_1, \ldots, \mu_k; \alpha_1, \ldots, \alpha_k}^{i_1, \ldots, i_k}\).

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

The Hamiltonian

Finally, the Hamiltonian is simply written as a sum of all possible terms

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

Note

The restrictions on the summation over the lattice and site indices are left undefined and considered to be a separate problem (see issue #2). The purpose of the general form is to provide a structure for writing down the parameter in a file. Said that, the general form implies three things

  • The cell (lattice vectors \(\mathbf{a}_1, \ldots, \mathbf{a}_d\)) and a set of sites in the cell (positions & labels) are defined file-wide.

  • The set of values for the cell and site indices (\(\mu_1, \ldots, \mu_k, \alpha_1, \ldots, \alpha_k\)) are defined for each individual parameter (for each tensor, not for each type of tensor).

  • The value of the constant \(C_k\), interpretation of each entity (position-wise), size of each component index are defined for a group of parameters (for example, parameters of the bilinear exchange, or parameters of the Zeeman term, etc.).

Mapping to the general form

The pages below discuss the mapping the terms present in the "zoo" to the general form of the Hamiltonian.