DMD - formalism

The central problem in applying the MF-RPA is the calculation of the dynamical susceptibility from equation (191). The standard procedure is to substitute equation (192) into (191), which is then solved for each desired value of and by a matrix inversion. In order to avoid a numerical divergence,it is necessary to add to a small imaginary constant and insert this into equation (192) leading to a susceptibility which is equivalent to equation (185). This method is inefficient and time demanding, however , because a matrix has to be inverted for each in the calculation.

In order to minimise the computational
effort an algorithm was developed [60], which requires only the solution
of a single generalised eigenvalue problem at each scattering vector .
This dynamical matrix diagonalisation (DMD) resembles the standard approach
to lattice dynamics.
This approach is very fast and will allow for more complex
systems to be investigated^{27}.

In the following we describe the DMD for a single excitation of each subsystem , i.e. we assume that each subsystem is a two level system with a single transition only. Other transitions (terms in equation (192)) can be considered in the DMD formalism by assigning to each of these transitions an additional value of the index and increasing the total number of subsystems () correspondingly. This procedure is different from adding other terms (which are present in equation (192)) to the right hand side of equation (194). However, both procedures lead to the same results, this is shown in [61].

For readability it is convenient to adopt the following matrix notation: a matrix is indicated by a bar on top of the symbol, e.g. refers to the matrix with . A matrix is denoted by a bar below the symbol. Making use of these two conventions the dynamical susceptibility can be written as .

Considering only a single excitation in the subsystem susceptibility, equation (192) can be rewritten as

with and the transition element matrix

Note for experts on programming single ion modules: that any external single ion module has to provide the matrix for every transition which is to be taken into consideration in the calculation. If the energy of this transition is zero, i.e. (diffuse scattering), the expression (195) would be zero because vanishes. In this case the single ion module should calculate instead of .

The matrices may be diagonalised giving eigenvalues which are all zero except for one real eigenvalue (which has the same sign as ):

(185) |

Now, the MF-RPA problem (190) may be simplified by using the unitary transformation ( ), which diagonalises . Note that the first column of this matrix (the eigenvector with the eigenvalue ) is simply

This property is useful as most of the equations below require
only knowledge of this first column. Following the
procedure outlined in [60] one may transform
the subsystem interaction

Now the Hermitian dynamical matrix may be defined as

(188) |

The energies of the system may be calculated by solving the following generalised eigenvalue problem:

where the matrix is defined as

The solution of the generalised eigenvalue problem (200) yields the eigenvectors and eigenvalues . These may be written as the eigenvalue matrix , and correspond to the excitation energies of the system at the wavevector for which was calculated. The solution of the eigenvalue problem (200) corresponds to the diagonalisation of the dynamical matrix in the case of phonons and therefore this method for calculating magnetic excitations is called dynamical matrix diagonalisation (DMD). Note, that the matrix does not change when the number of dimensions of the subsystem susceptibility is increased (for example to include quadrupolar degrees of freedom), unless there is an interaction coupling these degrees of freedom between different ions (this can be seen by the definition (198) of the matrix having in mind the first column of the transformation matrices , see equation (197)).

The eigenvector matrix provides a unitary transformation, which may be used to obtain the dynamical susceptibility. If the eigenvectors are normalised as , then equation (191) may be transformed using and (see [61]):

By its definition the generalised susceptibility gives information about the correlated
movement of the operators
for a specific excitation and contains the relative phases and amplitudes of the
different operators.
The procedure for the calculation of excitation energies and physical
observables (such as correlation functions and spectra) outlined above is very fast, because it
involves only a single diagonalisation (determination of the
matrix
) for every scattering
vector of interest. The
dynamical susceptibility does not need to be calculated for each
energy transfer by inverting equation
(190) saving much
computation time. Therefore the module *McDisp* of the *McPhase* package
uses this method by default. We want to emphasize, that the procedure outlined in this section is general
and allows to treat any number and combination of multipolar interactions just by
letting the index in
take values between 1 and the number
of multipolar operators considered (
).

Figure 26 illustrates the DMD algorithm. We have described the first three parts shown in the figure, obtaining the eigenvectors and eigenvalues . The following parts are described in the next section where the dynamical susceptibility is used to calculate a general susceptibility corresponding to an arbitrary observable, from which physical properties may be calculated.

Martin Rotter 2017-01-10