Dynamical Susceptibility and Excitations - Formalism for

This section is describes the formalism used in the calculation of the magnetic excitations. Because the procedure is not standard, we list the most important formulas.

We assume a quantum mechanical system that can be described by the Hamiltonian

The main limitation of the approach is that it neglects fluctuations associated with phase transitions and quantum disorder. We are primarily concerned, therefore, with excitations associated with a well-ordered ground state.

The translational symmetry of the system is represented by a Bravais lattice (which, in general, will be a superlattice of a crystal lattice). The position of subsystem can be specified by a lattice vector and a basis vector . The latter is the position of relative to . The index () labels the subsystems within the unit cell.

The calculation of the excited states of the
system starts from a mean-field model for the ground-state order.
We define a mean field acting on each subsystem by

The mean-field ground state is obtained from the self-consistent solution of (179) and (180). This iterative procedure is illustrated in fig. 25. The mean field Hamiltonian (180) for the subsystem is used to calculate the thermal expectation values for the initial mean field acting on all subsystems . Equation (179) is then used to calculate a new set of mean fields. These are again used in (180), and the procedure repeated until convergence is reached to within some specified precision. The free energy of the mean field ground state is evaluated and compared to that of other solutions obtained at the same temperature (computed from other initial states and superlattices). The solution with the lowest free energy corresponds to the stable ground state.

We now turn to the excited states. From linear response theory it can be shown [1, page 143] that the excited states are poles of the dynamical susceptibility, which is defined by

and

Here the energy levels and eigenstates
of the Hamiltonian (178) are denoted by and
, respectively.
is the corresponding
Boltzmann occupation probability.
and are
quantum mechanical operators describing the perturbation
to the Hamiltonian and the response of the system
according to the general concept of linear response theory [1].
The expression (181) is based on a system with well defined
energy levels implying that the poles of
are all lying on the real axis,
or that the absorptive part of the response function

The same result is obtained if is kept as a nonzero positive quantity in (181) instead of taking the limit , i.e. if assuming in the different terms in the sum and in the elastic term.

Because of the periodicity of our system we define generalized
susceptibilities
by choosing the Fourier transform operators

(175) | |||

(176) |

where is the number of unit cells. It will also be convenient to introduce the Fourier transform of the two-body interaction

(177) |

The calculation of the dynamical
susceptibility^{26}
from the Hamiltonian (5) is carried out
within the
mean field - random phase approximation
(MF-RPA) [1,59].
This approximation neglects correlations
in the differences
of
different subsystems .
In this approach the dynamical
susceptibility
for a primitive lattice ()
can be calculated from the solution to

where is the usual single ion magnetic susceptibility tensor. This equation can be written for the more general case of several subsystems ( ) as

or, in index notation, to

where for the sake of simplicity we omit the index on all quantities on the right-hand side. Here and are energy levels of the subsystem as calculated self-consistently within the mean-field theory using the Hamiltonian (180), and denote the corresponding eigenstates and the corresponding population numbers:

(182) |

The writing of (192) has been simplified in two ways. The obvious one is that should read where . Secondly, the elastic contribution is included in (192) by assuming the use of the following convention: is being replaced by in all terms where . The shift in energy introduced is and hence to leading order. Notice that the matrix elements of the thermal expectation values in (192) are only nonzero in the special cases of . Using the two conventions equation (192) becomes equivalent to (181) in the limit of (after taking the limit ). Since the expectation values are only needed in (192) when considering the elastic contribution, we may use this fact to signal that the second convention has to be applied whenever the expectation values are subtracted from the operators.

In order to evaluate
equations
(189)-(192) without producing a numerical divergence
it is necessary to add to a small imaginary constant
and insert this into equation (192).
If the option *-r * is used,
the program *McDisp* calculates the above expression for every energy
and stores the result in *./results/mcdisp.dsigma*.

If the option *-r* is not used, the
program *mcdisp* uses only the extremely fast DMD (Dynamical Matrix Diagonalisation)
algorithm[60] to calculate excitation energies and intensities and store the result in *mcdisp.qom,
mcdisp.qei, etc.*. The flowing chart of such a calculation is shown in fig. 26
and the formalism is outlined hereafter: