Attempt at generalization (#1)

Structure

Any 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 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.

• Two entities that are associated with two distinct sites commute.

  \[[X_{\mu_1; \alpha_1}^{i_1}, X_{\mu_2; \alpha_2}^{i_2}] = 0\]

  if \((\mu_1; \alpha_1) \neq (\mu_2; \alpha_2)\).

In the following whether the indices \((\mu_1; \alpha_1)\) and \((\mu_2; \alpha_2)\) are written within the same context it is implied that they indicate distinct sites: \((\mu_1; \alpha_1) \neq (\mu_2; \alpha_2)\).

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, whilst those entities are associates with one or more sites. 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 (cell index and site index) as the number of unique sites that it connects. For example, an interaction parameter

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

  connects \(m\) unique sites.

Note

Amount of entities and the amount of unique sites can be different. For example, an interaction parameter can connect two entities \(X_{\mu_1; \alpha_1}^{i_1}\) and \(X_{\mu_1; \alpha_1}^{i_2}\) is denoted as

\[V_{\mu_1; \alpha_1}^{i_1, i_2}\]

While an interaction parameter that connects two entities \(X_{\mu_1; \alpha_1}^{i_1}\) and \(X_{\mu_2; \alpha_2}^{i_2}\) is denoted as

\[V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}\]

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.

• Terms of the Hamiltonian can involve an arbitrary amount of unique sites.

• 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 \(m\) distinct sites

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

and \(k\) entities. Then the interaction parameter for such term is denoted as

\[V_{\mu_1, \ldots, \mu_m; \alpha_1, \ldots, \alpha_m}^{i_1, \ldots, i_k}\]

Now, the first question is - What are the restrictions on the values of \(k\) and \(m\)? The answer is

• \(k \ge m\).

  As every entity is associated with exactly one site.

• \(k \ne 0\) and \(m \ne 0\).

  If the Hamiltonian includes a term that is independent of the entities or does not involve any sites, then such a term either describes the environment and not the system or is a constant energy term that can be ignored.

With those rules in mind one can sketch a table (cross means that the term is not possible/considered) for one possible classification of the Hamiltonian's terms.

	\(m = 1\)	\(m = 2\)	\(m = 3\)	\(m = 4\)	\(\ldots\)
\(k = 1\)		\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 2\)			\(\times\)	\(\times\)	\(\times\)
\(k = 3\)				\(\times\)	\(\times\)
\(k = 4\)					\(\times\)
\(\ldots\)

Unfortunately, one can not simply classify the terms of the Hamiltonian by the pair \((k, m)\). First counter-example occurs when \(k = 4\). Imagine two terms that both involve \(k = 4\) entities and \(m = 2\) sites

\[V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} X_{\mu_1; \alpha_1}^{i_1} X_{\mu_1; \alpha_1}^{i_2} X_{\mu_2; \alpha_2}^{i_3} X_{\mu_2; \alpha_2}^{i_4} \qquad \text{and} \qquad V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} X_{\mu_1; \alpha_1}^{i_1} X_{\mu_1; \alpha_1}^{i_2} X_{\mu_1; \alpha_1}^{i_3} X_{\mu_2; \alpha_2}^{i_4}\]

Those two terms are distinct and can not be express one through another. Therefore, the label \((k, m)\) is not sufficient to uniquely identify a term of the Hamiltonian.

Hint

On the other, hand the terms

\[ \begin{align}\begin{aligned}\begin{split}&V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} X_{\mu_1; \alpha_1}^{i_1} X_{\mu_1; \alpha_1}^{i_2} X_{\mu_1; \alpha_1}^{i_3} X_{\mu_2; \alpha_2}^{i_4} \\\end{split}\\\begin{split}&V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} X_{\mu_1; \alpha_1}^{i_1} X_{\mu_1; \alpha_1}^{i_2} X_{\mu_2; \alpha_2}^{i_3} X_{\mu_1; \alpha_1}^{i_4} \\\end{split}\\\begin{split}&V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} X_{\mu_1; \alpha_1}^{i_1} X_{\mu_2; \alpha_2}^{i_2} X_{\mu_1; \alpha_1}^{i_3} X_{\mu_1; \alpha_1}^{i_4} \\\end{split}\\\begin{split}&V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} X_{\mu_2; \alpha_2}^{i_1} X_{\mu_1; \alpha_1}^{i_2} X_{\mu_1; \alpha_1}^{i_3} X_{\mu_1; \alpha_1}^{i_4} \\\end{split}\end{aligned}\end{align} \]

are equivalent, as two entities associated with two distinct sites commute (therefore, all of those four terms can be expressed through the first one by re-ordering of the entities and renaming of the indices).

Now the questions are

• How many distinct terms of the Hamiltonian exist for each \(k\)?

• How those can be labeled?

To answer those, define a set of numbers \(\{n_1, n_2, \ldots, n_m\}\), where each number \(n_j\) indicates how many entities are associated with the site \((\mu_j; \alpha_j)\). As the total amount of entities is \(k\), one gets

\[\sum_j n_j = k\]

Then, the first question can be re-formulated as

• How many integer partitions of an integer \(k\) exist (see Integer partition (Wikipedia))?

The answer is a partition function, that is labeled as \(p(k)\). Here are its values for a few first values of \(k\)

\(k\)	1	2	3	4	5	6	\(\ldots\)
\(p(k)\)	1	2	3	5	7	11	\(\ldots\)

Therefore, the terms of the Hamiltonian can be uniquely labeled by a pair \((k, l)\), where \(k\) is the amount of entities and \(l\) is an index of a particular integer partition of the number \(k\) (with \(l = 1, \ldots, p(k)\)). For each such pair define a number \(m_{k,l}\), which indicates how many unique sites are involved in the term \((k, l)\).

Hint

To be exact, pick the ordering on the partitions of an integer \(k\) to be an descending order of sets of integers that specified a partition, while each set is ordered in a descending order. Here is an example of such ordering

• \(k = 1\)

  Partition	\(l\)
  1	1

• \(k = 2\)

  Partition	\(l\)
  2	1
  1 + 1	2

• \(k = 3\)

  Partition	\(l\)
  3	1
  2 + 1	2
  1 + 1 + 1	3

• \(k = 4\)

  Partition	\(l\)
  4	1
  3 + 1	2
  2 + 2	3
  2 + 1 + 1	4
  1 + 1 + 1 + 1	5

Redesign a classification table by using the labels \((k, l)\) (instead of \((k, m)\)). As entries of the table put the amount of distinct sites \(m_{k,l}\) that are involved in the term.

	\(l = 1\)	\(l = 2\)	\(l = 3\)	\(l = 4\)	\(l = 5\)	\(\ldots\)
\(k = 1\)	1	\(\times\)	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 2\)	1	2	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 3\)	1	2	3	\(\times\)	\(\times\)	\(\times\)
\(k = 4\)	1	2	2	3	4	\(\times\)
\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)

or the repetition numbers \(\{n_{l, 1}, \ldots, n_{l, m_{k,l}}\}\)

	\(l = 1\)	\(l = 2\)	\(l = 3\)	\(l = 4\)	\(l = 5\)	\(\ldots\)
\(k = 1\)	1	\(\times\)	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 2\)	2	1, 1	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 3\)	3	2, 1	1, 1, 1	\(\times\)	\(\times\)	\(\times\)
\(k = 4\)	4	3, 1	2, 2	2, 1, 1	1, 1, 1, 1	\(\times\)
\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)

The Hamiltonian

Finally, everything is ready to write the Hamiltonian.

The Hamiltonian is a sum over \(k\) and over \(l\)

\[\mathcal{H} = \sum_{k=1}^{\infty} \sum_{l=1}^{p(k)} \mathcal{H}_{k, l}\]

Before introducing the general form of \(\mathcal{H}_{k, l}\), consider a few examples.

Hint

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

k = 1

One case is possible.

• \(l = 1\): \(m_{1,1} = 1\), \(n_1 = 1\)

  \[\begin{split}\mathcal{H}_{1, 1} = C_{1, 1} \sum_{\substack{\mu_1, \\ \alpha_1, \\ i_1}} V_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_1}\end{split}\]

k = 2

Two cases are possible.

• \(l = 1\): \(m_{2,1} = 1\), \(n_1 = 2\)

  \[\begin{split}\mathcal{H}_{2, 1} = C_{2, 1} \sum_{\substack{\mu_1, \\ \alpha_1, \\ i_1, i_2}} V_{\mu_1; \alpha_1}^{i_1, i_2} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_2}\end{split}\]

• \(l = 2\): \(m_{2,2} = 2\), \(n_1 = 1\), \(n_2 = 1\)

  \[\begin{split}\mathcal{H}_{2, 2} = C_{2, 2} \sum_{\substack{\mu_1, \mu_2, \\ \alpha_1, \alpha_2, \\ i_1, i_2}} V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_2; \alpha_2}^{i_2}\end{split}\]

k = 3

Three cases are possible.

• \(l = 1\): \(m_{3,1} = 1\), \(n_1 = 3\)

  \[\begin{split}\mathcal{H}_{3, 1} = C_{3, 1} \sum_{\substack{\mu_1, \\ \alpha_1, \\ i_1, i_2, i_3}} V_{\mu_1; \alpha_1}^{i_1, i_2, i_3} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_2} \cdot X_{\mu_1; \alpha_1}^{i_3}\end{split}\]

• \(l = 2\): \(m_{3,2} = 2\), \(n_1 = 2\), \(n_2 = 1\)

  \[\begin{split}\mathcal{H}_{3, 2} = C_{3, 2} \sum_{\substack{\mu_1, \mu_2, \\ \alpha_1, \alpha_2, \\ i_1, i_2, i_3}} V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_2; \alpha_2}^{i_2} \cdot X_{\mu_2; \alpha_2}^{i_3}\end{split}\]

• \(l = 3\): \(m_{3,3} = 3\), \(n_1 = 1\), \(n_2 = 1\), \(n_3 = 1\)

  \[\begin{split}\mathcal{H}_{3, 3} = C_{3, 3} \sum_{\substack{\mu_1, \mu_2, \mu_3, \\ \alpha_1, \alpha_2, \alpha_3, \\ i_1, i_2, i_3}} V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_2; \alpha_2}^{i_2} \cdot X_{\mu_3; \alpha_3}^{i_3}\end{split}\]

k = 4

Five cases are possible.

• \(l = 1\): \(m_{4,1} = 1\), \(n_1 = 4\)

  \[\begin{split}\mathcal{H}_{4, 1} = C_{4, 1} \sum_{\substack{\mu_1, \\ \alpha_1, \\ i_1, i_2, i_3, i_4}} V_{\mu_1; \alpha_1}^{i_1, i_2, i_3, i_4} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_2} \cdot X_{\mu_1; \alpha_1}^{i_3} \cdot X_{\mu_1; \alpha_1}^{i_4}\end{split}\]

• \(l = 2\): \(m_{4,2} = 2\), \(n_1 = 3\), \(n_2 = 1\)

  \[\begin{split}\mathcal{H}_{4, 2} = C_{4, 2} \sum_{\substack{\mu_1, \mu_2, \\ \alpha_1, \alpha_2, \\ i_1, i_2, i_3, i_4}} V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_2} \cdot X_{\mu_1; \alpha_1}^{i_3} \cdot X_{\mu_2; \alpha_2}^{i_4}\end{split}\]

• \(l = 3\): \(m_{4,3} = 2\), \(n_1 = 2\), \(n_2 = 2\)

  \[\begin{split}\mathcal{H}_{4, 3} = C_{4, 3} \sum_{\substack{\mu_1, \mu_2, \\ \alpha_1, \alpha_2, \\ i_1, i_2, i_3, i_4}} V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2, i_3, i_4} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_2} \cdot X_{\mu_2; \alpha_2}^{i_3} \cdot X_{\mu_2; \alpha_2}^{i_4}\end{split}\]

• \(l = 4\): \(m_{4,4} = 3\), \(n_1 = 2\), \(n_2 = 1\), \(n_3 = 1\)

  \[\begin{split}\mathcal{H}_{4, 4} = C_{4, 4} \sum_{\substack{\mu_1, \mu_2, \mu_3, \\ \alpha_1, \alpha_2, \alpha_3, \\ i_1, i_2, i_3, i_4}} V_{\mu_1, \mu_2, \mu_3; \alpha_1, \alpha_2, \alpha_3}^{i_1, i_2, i_3, i_4} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_1; \alpha_1}^{i_2} \cdot X_{\mu_2; \alpha_2}^{i_3} \cdot X_{\mu_3; \alpha_3}^{i_4}\end{split}\]

• \(l = 5\): \(m_{4,5} = 4\), \(n_1 = 1\), \(n_2 = 1\), \(n_3 = 1\), \(n_4 = 1\)

  \[\begin{split}\mathcal{H}_{4, 5} = C_{4, 5} \sum_{\substack{\mu_1, \mu_2, \mu_3, \mu_4, \\ \alpha_1, \alpha_2, \alpha_3, \alpha_4, \\ i_1, i_2, i_3, i_4}} V_{\mu_1, \mu_2, \mu_3, \mu_4; \alpha_1, \alpha_2, \alpha_3, \alpha_4}^{i_1, i_2, i_3, i_4} \cdot X_{\mu_1; \alpha_1}^{i_1} \cdot X_{\mu_2; \alpha_2}^{i_2} \cdot X_{\mu_3; \alpha_3}^{i_3} \cdot X_{\mu_4; \alpha_4}^{i_4}\end{split}\]

Arbitrary k

Generalizing the examples above, one can write the general form of \(\mathcal{H}_{k, l}\) as

\[\begin{split}\mathcal{H}_{k, l} = C_{k, l} \sum_{\substack{\mu_1, \ldots, \mu_{m_{k,l}}, \\ \alpha_1, \ldots, \alpha_{m_{k,l}}, \\ i_1, \ldots, i_k}} V_{\mu_1, \ldots, \mu_{m_{k,l}}; \alpha_1, \ldots, \alpha_{m_{k,l}}}^{i_1, \ldots, i_k} \cdot \Biggl( X_{\mu_1; \alpha_1}^{i_1} \cdot \ldots \cdot X_{\mu_1; \alpha_1}^{i_{n_{l,1}}} \Biggr) \cdot \ldots \cdot \Biggl( X_{\mu_{m_{k,l}}; \alpha_{m_{k,l}}}^{i_{k - n_{l,m_{k,l}} + 1}} \cdot \ldots \cdot X_{\mu_{m_{k,l}}; \alpha_{m_{k,l}}}^{i_k} \Biggr)\end{split}\]

where \(n_{l, j}\) indicates how many entities are associated with site \((\mu_j; \alpha_j)\) in the integer partition \(l\) of the number \(k\) and \(m_{k,l}\) is the amount of different sites involved in the term that corresponds to the integer partition \(l\) of the number \(k\).

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.

• Bilinear exchange
• Biquadratic exchange
• Crystal field
• Magnetic dipole-dipole interaction
• Exchange striction
• Magneto-elastic interaction
• On-site anisotropy (k = 2)
• On-site anisotropy (k = 4)
• Quadruplet interaction
• Two-ion crystal field
• Zeeman interaction

• On-site anisotropy (k)

The terms of the Hamiltonian, that are discussed above, can be classified as follows

	\(l = 1\)	\(l = 2\)	\(l = 3\)	\(l = 4\)	\(l = 5\)	\(\ldots\)
\(k = 1\)	Crystal field, Crystal field (ambiguity), Magneto-elastic interaction (1, 1), Zeeman interaction	\(\times\)	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 2\)	Crystal field (ambiguity), On-site anisotropy (k = 2)	Bilinear exchange, Magnetic dipole-dipole interaction, Exchange striction, Two-ion crystal field, Two-ion crystal field (ambiguity), Magneto-elastic interaction (2, 2),	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 3\)	Crystal field (ambiguity)	Two-ion crystal field (ambiguity)	3	\(\times\)	\(\times\)	\(\times\)
\(k = 4\)	Crystal field (ambiguity), On-site anisotropy (k = 4)	Two-ion crystal field (ambiguity)	Biquadratic exchange	Quadruplet interaction (case 1)	Quadruplet interaction (case 2)	\(\times\)
\(\ldots\)	Crystal field (ambiguity)	Two-ion crystal field (ambiguity)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)

The table below gives an idea of which terms are implemented in each code (\(^*\) indicates that the entry is due to the ambiguity of crystal field parameterization, see Crystal field (ambiguity) and Two-ion crystal field (ambiguity))

	\(l = 1\)	\(l = 2\)	\(l = 3\)	\(l = 4\)	\(l = 5\)	\(\ldots\)
\(k = 1\)	Magnopy, Magpie, McPhase, SpinW Spirit Sunny	\(\times\)	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 2\)	GROGU, juKKR, Magnopy, Magpie, McPhase\(^*\), SpinW, Spirit, Sunny\(^*\), TB2J	GROGU, juKKR, Magnopy, Magpie, McPhase, SpinW, Spirit, Sunny, TB2J	\(\times\)	\(\times\)	\(\times\)	\(\times\)
\(k = 3\)	Magnopy, McPhase\(^*\), Sunny\(^*\)	Magnopy, McPhase\(^*\)	Magnopy	\(\times\)	\(\times\)	\(\times\)
\(k = 4\)	Magnopy, McPhase\(^*\), Spirit, Sunny\(^*\)	Magnopy, McPhase\(^*\)	Magnopy, SpinW, Spirit	Magnopy, Spirit	Magnopy, Spirit	\(\times\)
\(\ldots\)	McPhase\(^*\), Sunny\(^*\)	McPhase\(^*\)	\(\ldots\)	\(\ldots\)	\(\ldots\)	\(\ldots\)