Chi scaling analysis

This article: Measuring multipartite entanglement through dynamic susceptibilities proposed a scaling relation for the susceptibility.

$$\chi^{\prime\prime}(\omega, T) \sim T^{-s} \phi(\omega/ T)$$

where $\chi^{\prime\prime}$ is the imaginary part of the susceptibility, $T$ is the temperature, $s$ an exponent, and $\phi$ is a universal function independent of the system.

In Finite-temperature scaling of spin correlations in a partially magnetized Heisenberg S=1/2 chain, the authors study the spin susceptibility and find a scaling relation with the scaling function in the form of

$$\phi(x) \sim \mathrm{Im}\left[\left(\frac{\Gamma(1/(8K) - ix/(4\pi))\,\Gamma(1-1/(4K))}{\Gamma(1-1/(8K) - i x / (4\pi))}\right)^2\right]$$

with $s = 2-1/(2K)$. The neutron scattering data seem to follow this scaling relation.

To apply this scaling relation to our data, we need the charge susceptibility and figure out the charge-specific universal function $\phi$. In Fourier transform of the 2kF Luttinger liquid density correlation function with different spin and charge velocities, the retarded charge density susceptibility of Luttinger liquid is theoretically derived,

$$\chi^{\prime\prime}(\omega, T) \sim T^{K-2} \times \underbrace{\mathrm{Im}\left[B\!\left(-\frac{i\omega}{T} + \frac{K}{4},\, 1-\frac{K}{2}\right)^2\right]}_{\phi}$$

They claim that $K = 1$.

The model we use: $$\chi^{\prime\prime}(\omega, T) \sim T^{K-2} \phi(\omega / T)$$

We select the energy window of $\omega \in (20, 40)\,\mathrm{meV}$ and use only these data for the analysis. We also assume that the RIXS intensity of the charge excitations is proportional to the dynamical structure factor:

$$I_\text{RIXS} \propto S(q, \omega) \sim \frac{1}{1-e^{-\omega/T}} \chi^{\prime\prime}(q, \omega)$$

Here are the steps:

1. For each temperature, we calculate the susceptibility $\chi^{\prime\prime}(\omega, T)$ by multiplying the RIXS intensity with the Bose factor $1-e^{-\omega/T}$:

$$\chi^{\prime\prime}(\omega, T) \sim \left(1-e^{-\omega/T}\right) I_\text{RIXS}(\omega, T)$$

1. Crop the data and only keep those lying within the energy window of $\omega \in (20, 40)\,\mathrm{meV}$.

2. Plot the data with the $y$-axis rescaled by $T^{K-2}$ and the $x$-axis rescaled by $T$, both in log scale.

3. Fit the data with the scaling function $\phi$ and extract the value of $K$.

It turns out that we also found $K \sim 1$, consistent with the theoretical prediction for the charge sector of a Luttinger liquid.

Here we integrate the $\chi^{\prime\prime}$ over the full energy range, weight by $\tanh(\omega/2T)$ to get the Fisher information $F_Q$.

$$F_Q(T) \equiv \int_{0}^{100\mathrm{meV}} \, d\omega \tanh\left(\frac{\omega}{2T}\right) \chi^{\prime\prime}(\omega, T)$$

Then we use a power-law function $y \propto T^{K-1}$ to fit the data. It turned out that $K\sim 0.74$.

In principle, if the susceptibility strictly follows the scaling relation $\chi^{\prime\prime} \sim T^{K-2}\phi(\omega/T)$, the integration of it will scale as $F_Q \sim T^{K-1}$ with the same $K$. We got different scaling parameter $K$: susceptibility scales with $K\sim 1$ and Fisher information scales with $K\sim 0.74$.

This is not surprising because in our case $\chi^{\prime\prime}$ does not collapse to a single curve. It only works if we select specific segment of the data (20-40 meV).

The analysis is implemented in susceptibility/scaling_analysis.py.

Data flow:

1. Load data/processed_exp.pkl.
2. Extract RIXS intensity at $q = 0.24$ r.l.u. from exp.Interpolated_subtracted_realigned_interesting_data.
3. Convert intensity to $\chi''$ via the Bose factor.
4. Fit the scaling collapse with scipy.optimize.curve_fit.
5. Save figures under susceptibility/figures/.