QFI normalized by elastic

First we show the elastic intensity as a function of momentum $H$ at four temperatures.

We fit the elastic intensity profile (for each temperature) with a Lorentzian + constant model in momentum, using data with $H \ge 0.15$:

$$I_{\mathrm{elastic}}=\frac{A\,\gamma^2}{(H-H_0)^2+\gamma^2}+const,$$

with $\mathrm{FWHM}=2\gamma$.

The figure below summarizes the fitted quantities as a function of temperature.

The unnormalized QFI-like quantity:

$$F_Q(T,H)=\int_{0}^{\omega_{\max}} I(T,H,\omega)\,\tanh\!\left(\frac{\omega}{2T_{\mathrm{meV}}}\right)d\omega$$

where $I$ is the RIXS intensity with elastic subtracted, including phonon contributions.

Here we normalize $F_Q$ by the $E_{\mathrm{norm}}(21\,\mathrm{K})$ (defined below) computed at $T = 21\,\mathrm{K}$ only, and apply this single constant to all temperatures:

$$\widetilde{F}_Q(T,H)=\frac{F_Q(T,H)}{E_{\mathrm{norm}}(21\,\mathrm{K})},$$

where

$$E_{\mathrm{norm}}(21\,\mathrm{K})=A(21\,\mathrm{K})\cdot\Delta\omega_{\mathrm{el}},$$

with $A$ the elastic amplitude and $\Delta\omega_{\mathrm{el}} = 22.5\,\mathrm{meV}$ the instrumental energy resolution. Multiplying by $\Delta\omega_{\mathrm{el}}$ ensures $\widetilde{F}_Q$ is dimensionless, since $F_Q$ carries units of intensity$\times$meV.

Here the normalization is temperature-dependent, and the elastic norm now includes both the momentum linewidth and the energy resolution:

$$E_{\mathrm{norm}}(T)=A(T)\cdot\mathrm{FWHM}(T)^2\cdot\Delta\omega_{\mathrm{el}},$$

where $A(T)$ is the elastic amplitude, $\mathrm{FWHM}(T)$ is the full-width at half-maximum of the elastic peak in momentum (r.l.u.), and $\Delta\omega_{\mathrm{el}} = 22.5\,\mathrm{meV}$ is the instrumental energy FWHM. The normalized quantity is:

$$\widetilde{F}_Q(T,H)=\frac{F_Q(T,H)}{E_{\mathrm{norm}}(T)}.$$

This normalization is motivated by the Phys. Rev. Lett. 124, 187002 (2020), where elastic intensity is reported to be proportional to the square of the correlation length $\xi$, i.e.\ $A\cdot\mathrm{FWHM}^2 \sim A/\xi^2$ is expected to be temperature- and material-independent.

I currently do not have a good idea how to get the absolute value of the susceptibility, and therefore I normalize it to the elastic. According to PHYSICAL REVIEW X 6, 041019 (2016), the RIXS intensity of the charge excitations is (in the most simplified form):

$$I_\mathrm{charge} \sim \sum_p \frac{f(\epsilon_{p+q}) - f(\epsilon_{p})}{|\epsilon_{p} - \epsilon_{p+q} + i\Gamma + \omega|^2}$$

The square in the denominator, if I understand correctly, comes from the fact that RIXS is a second-order process. While on the other hand, if we use Lindhard function to approximate the density-density susceptibility, it would be

$$\chi_\mathrm{charge} \sim \sum_p \frac{f(\epsilon_{p+q}) - f(\epsilon_{p})}{\epsilon_{p} - \epsilon_{p+q} + i\Gamma + \omega}$$

The difference is in the denominator.

Therefore, I don't know how I can directly get the absolute value of the susceptibility from the RIXS intensity. At the moment the best I can do is to normalize it to the (pseudo-)elastic intensity, hoping for another way to circumvent this problem. At least now the \(F_Q\) is dimensionless.

This report follows the existing standalone workflow in susceptibility/:

• reproduce_figure4a.py: generates the four-temperature intensity maps.
• figure4a_qfi_1d.py: generates QFI-weighted 1D cuts with the $\tanh(\omega/2T)$ overlay.
• figure4a_qfi_integration_energy.py: computes the original QFI-like energy integration.
• normalize_to_elastic.py: fits elastic intensity vs momentum and defines the elastic norm.
• figure4a_qfi_integration_energy_normalized_elastic.py: computes the new elastic-normalized integral.

All figures are saved under susceptibility/figures/.