Calculation of the Correlation Function of Physical Observables

Physical observables are related to operators such as the magnetic moment components or the scattering operators. We now introduce a series of observables $\hat \mathcal O_{\alpha}$ ( $\alpha=1,...,m'$) for each subsystem $n=(\ensuremath{\boldsymbol\ell}s)$ and define a corresponding correlation function and dynamical susceptibility. The correlation function we denote as $\underline{\overline{\Sigma}}({\mathbf Q},\omega)$

$${\Sigma^{ss'}_{\alpha\beta}}({\mathbf Q},\omega) = \int_{-\infty}^{+\infty}dt e^{i\omega t}e^{-i\mathbf Q \cdot(\mat... ...hbf Q \cdot(\ensuremath{\boldsymbol\ell}-\ensuremath{\boldsymbol\ell}')} \times$$

$$\times(\langle \hat \mathcal O^{\ensuremath{\boldsymbol\ell}s \da... ...gle \hat \mathcal O^{\ensuremath{\boldsymbol\ell}' s'}_{\beta} % \rangle_{T,H})$$ (225)

Note that the observables $\hat \mathcal O_{\alpha}^{\ensuremath{\boldsymbol\ell}s}$ may have some $\mathbf Q$ dependence (for example if we consider the Fourier transform of a magnetisation density of a magnetic ion in a crystal [49]). We omit this $\mathbf Q$ dependence to make notation easier but will come back to it when considering the calculation of the neutron scattering cross section.

Using the fluctuation-dissipation theorem $\Sigma$ can be related to a dynamical susceptibility $X$:

$${\underline{\overline{\Sigma}}}({\mathbf Q},\omega) = \frac{... ...ft ( {\underline{\overline{X}}} \right )''({\mathbf Q},\omega)$$ (226)

$$\left (X_{\alpha\beta}^{ss'}\right )''({\mathbf Q},\omega)\e... ...alpha\beta}^{ss'}({\mathbf Q},\omega) \right )^{\ast}} \right]$$ (227)

However, in contrast to the dynamical susceptibility $\underline{\overline{\chi}}$, based on the operators $\hat \mathcal I_{\alpha}$, the susceptibility $\underline{\overline{X}}$ cannot be obtained from solving the MF-RPA equation (226). The reason is that the derivation of (226) makes use of the dynamical evolution of the operators $\hat \mathcal I_{\alpha}$ and not of the observables $\hat \mathcal O_{\alpha}$.

Nonetheless, general properties of dynamical susceptibilities may be used to get a relation between $\overline{\underline{X}}({\mathbf Q},\omega)$ and $\overline{\underline{\chi}}({\mathbf Q},\omega)$. In general a dynamical susceptibility $\chi_{BA}(\omega)$ describes the response of a physical observable $\langle\hat B(t)\rangle$ to a perturbation of the system described by an operator $\hat A$. In the case of the dynamical susceptibility corresponding to the observables $\mathcal O_{\alpha}$ we therefore have to set $X_{\alpha\beta}^{ss'}({\mathbf Q},\omega)\equiv\chi_{DC}(\omega)$ with the definitions

$$\hat C=\frac{1}{\sqrt{N_g}}e^{i\mathbf Q \cdot\mathbf b_{s'}}\sum... ...f Q \cdot\ensuremath{\boldsymbol\ell}'}\hat \mathcal O^{\mathbf % l's'}_{\beta}$$ (228)

$$\hat D=\frac{1}{\sqrt{N_g}}e^{i\mathbf Q \cdot\mathbf b_{s}}\sum_... ...hbf Q \cdot\ensuremath{\boldsymbol\ell}}\hat \mathcal O^{\mathbf % ls}_{\alpha}$$ (229)

From linear response theory it can be shown [1, page 143], that the dynamical susceptibilities have poles at the excitation energies of the system: In equation (216) the denominator, the eigenstates and the difference in thermal population are the same for any susceptibility. The energy eigenstates $\vert\alpha \rangle$ of the system will be a linear combination of direct products involving single ion states. Therefore, the numerator in equation (216) will be a (usually not known) linear combination of products of the form $\langle -\vert\hat \mathcal I_{\alpha}^s\vert+\rangle \langle +'\vert\hat \mathcal I_{\beta}^{s'}\vert-'\rangle$ and $\langle -\vert\hat \mathcal O_{\alpha}^s\vert+\rangle \langle +'\vert\hat \mathcal O_{\beta}^{s'}\vert-'\rangle$ for $\chi_{\alpha\beta}^{ss'}({\mathbf Q},\omega)$ and $X_{\alpha\beta}^{ss'}({\mathbf Q},\omega)$, respectively.

These terms can be related by a similar procedure to that outlined in equations (230) ff. We define the matrices,

$$\mu^s_{\alpha\beta}=\langle-\vert\hat \mathcal I^{s}_{\alpha}-\la... ...\hat \mathcal I^s_{\beta}-\langle \hat \mathcal I^s_{\beta}\rangle\vert-\rangle$$ (230)

$$\nu^s_{\alpha\beta}=\langle-\vert\mathcal O^{s \dag }_{\alpha}-\l... ...ngle+\vert\mathcal O^s_{\beta}-\langle \mathcal O^s_{\beta}\rangle\vert-\rangle$$ (231)

similar to the $\overline{M}^s$ of equation (230), however, omitting the thermal population factors and expectation values. These matrices can be diagonalised using the unitary transformations $\overline{\mathcal U}^s$ ( $\overline{\mathcal U}^{s\dag }\overline{\mathcal U}^s=\overline{1}$) and $\overline{\mathcal V}^s$ ( $\overline{\mathcal V}^{s\dag }\overline{\mathcal V}^s=\overline{1}$), respectively.

Again, all eigenvalues are zero except for $\phi^s$ and $\xi^s$, respectively.

$$\phi^s \equiv \gamma ^s/(p_--p_+)={\rm Trace \{\overline{\mu^s} \} }$$ (232)

$$\xi^s \equiv \Gamma^s /(p_--p_+) ={\rm Trace \{\overline{\nu^s} \} }$$ (233)

In [35] these transformations are used to derive the following expression for the dynamical susceptibility

$$X^{ss'}_{\alpha\beta}=(\sqrt{\Gamma^s})^\ast \sum_r \mathca... ...T^{rs'\dag }} \mathcal V^{s'\dag }_{1\beta} \sqrt{\Gamma^{s'}}$$ (234)

Equation (247) shows how knowledge of the dynamical susceptibility calculated on the basis of the interaction operators between subsystems ( $\hat \mathcal I_{\alpha}^s$) may be used to obtain the dynamical susceptibility for any set of observables $\hat \mathcal O_{\alpha}$ of the system.

The standard procedure to avoid divergences is to substitute $\hbar\omega$ with $\hbar \omega + i \hbar \varepsilon$ and take the limit for $\varepsilon \rightarrow 0^+$. Using Dirac's formula

$$\lim_{ \varepsilon \rightarrow 0^+} \frac{1}{\hbar {\omega^... ... \hbar \omega} + i \pi \delta(\hbar {\omega^r} -\hbar \omega)$$ (235)

the absorptive part of the dynamical susceptibility (240) becomes

$$\left (X^{ss'}_{\alpha\beta}\right )''=\pi(\sqrt{\Gamma^s})^... ...}(\mathbf Q) \mathcal V^{s'\dag }_{1\beta} \sqrt{\Gamma^{s'}}$$ (236)

and the correlation function $\underline{\overline{\Sigma}}$ can be evaluated by applying the fluctuation dissipation theorem (239):

$${\Sigma^{ss'}_{\alpha\beta}({\mathbf Q},\omega)= \frac{2 \p... ...r \omega) \mathcal T^{rs'\dag } \mathcal V^{s'\dag }_{1\beta}}$$ (237)

To keep notation simple, the $\mathbf Q$ dependence of the eigenvectors $\underline{\mathcal T}$ and the energies $\omega^r$ has been omitted. If the observable $\hat \mathcal O_{\alpha}$ depends explicitly on $\mathbf Q$, then also $\overline{\nu}^s$ and thus $\overline{\mathcal V}^s$ will depend on $\mathbf Q$. This very fundamental result will be applied to the neutron scattering cross section in section M.

The elastic contribution to equations (249) and (250) has to be evaluated taking into account a small but finite value for the energy shift $d$ introduced in the discussion of equation (227). It turns out, that $\gamma^s$ and $\Gamma^s$ and thus also the dynamical matrix $\underline{A}$ and its eigenvalues are proportional to $d$. Making use of the normalisation for the eigenvectors $\underline{\mathcal T}^\dag\underline{A}(\mathbf Q)\underline{\mathcal T}=\underline{1}$ we find that the dynamical susceptibility (249) is proportional to $d$ and thus zero in the limit of $d \rightarrow 0$. However, in the correlation function (250) the denominator is proportional to $d$ leading to a finite result for the quasielastic response.