Model-Independent Sum Rule Analysis Based on Limited-Range Spectral Data

TL;DR

This paper introduces a model-independent method to accurately determine spectral sum rules within limited spectral data ranges using Kramers-Kronig relations, eliminating the need for risky extrapolations.

Contribution

It demonstrates that sum-rule integrals can be derived directly from limited spectral data without additional model assumptions, enhancing experimental analysis reliability.

Findings

Sum rules can be obtained without extrapolation.

Applicable to various response functions like optical conductivity.

Enables accurate spectral weight determination at low frequencies.

Abstract

Partial sum rules are widely used in physics to separate low- and high-energy degrees of freedom of complex dynamical systems. Their application, though, is challenged in practice by the always finite spectrometer bandwidth and is often performed using risky model-dependent extrapolations. We show that, given spectra of the real and imaginary parts of any causal frequency-dependent response function (for example, optical conductivity, magnetic susceptibility, acoustical impedance etc.) in a limited range, the sum-rule integral from zero to a certain cutoff frequency inside this range can be safely derived using only the Kramers-Kronig dispersion relations without any extra model assumptions. This implies that experimental techniques providing both active and reactive response components independently, such as spectroscopic ellipsometry in optics, allow an extrapolation-independent determination of spectral weight 'hidden' below the lowest accessible frequency.