Analytical Continuation of Matrix-Valued Functions: Carath\'eodory Formalism

TL;DR

This paper develops a mathematical framework and an interpolation algorithm for the analytic continuation of matrix-valued Green's functions in quantum field theories, enabling more accurate real-frequency analysis of multi-orbital systems.

Contribution

It introduces criteria for the existence of such functions and presents an algorithm that respects their mathematical properties, improving analytic continuation methods.

Findings

Exact recovery of off-diagonal and diagonal elements in small systems with precise data.

Sufficient precision for continuation to commute with Dyson equation in real-materials.

Highlighting artifacts caused by truncating off-diagonal self-energy elements.

Abstract

Finite-temperature quantum field theories are formulated in terms of Green's functions and self-energies on the Matsubara axis. In multi-orbital systems, these quantities are related to positive semidefinite matrix-valued functions of the Carath\'eodory and Schur class. Analysis, interpretation and evaluation of derived quantities such as real-frequency response functions requires analytic continuation of the off-diagonal elements to the real axis. We derive the criteria under which such functions exist for given Matsubara data and present an interpolation algorithm that intrinsically respects their mathematical properties. For small systems with precise Matsubara data, we find that the continuation exactly recovers all off-diagonal and diagonal elements. In real-materials systems, we show that the precision of the continuation is sufficient for the analytic continuation to commute with the Dyson equation, and we show that the commonly used truncation of off-diagonal self-energy elements leads to considerable approximation artifacts. Our method paves the way for the systematic evaluation of Matsubara data with equations of many-body theory on the real-frequency axis.