# Maximum entropy formalism for the analytic continuation of matrix-valued Green's functions

## TL;DR

This paper extends the maximum entropy method to analytically continue matrix-valued Green's functions, addressing off-diagonal elements and physical constraints, with applications to DMFT and real materials like LaTiO3.

## Contribution

The paper introduces a generalized maximum entropy approach for matrix Green's functions, including a computationally efficient element-wise method and a full matrix formalism ensuring physical constraints.

## Key findings

- Element-wise method effectively continues matrix Green's functions.
- Full matrix formalism preserves positive semidefiniteness and Hermiticity.
- Application to LaTiO3 demonstrates practical relevance.

## Abstract

We present a generalization of the maximum entropy method to the analytic continuation of matrix-valued Green's functions. To treat off-diagonal elements correctly based on Bayesian probability theory, the entropy term has to be extended for spectral functions that are possibly negative in some frequency ranges. In that way, all matrix elements of the Green's function matrix can be analytically continued; we introduce a computationally cheap element-wise method for this purpose. However, this method cannot ensure important constraints on the mathematical properties of the resulting spectral functions, namely positive semidefiniteness and Hermiticity. To improve on this, we present a full matrix formalism, where all matrix elements are treated simultaneously. We show the capabilities of these methods using insulating and metallic dynamical mean-field theory (DMFT) Green's functions as test cases. Finally, we apply the methods to realistic material calculations for LaTiO$_3$, where off-diagonal matrix elements in the Green's function appear due to the distorted crystal structure.