Validity and Limits of Low Order Hybridization Expansion Approaches for Multi-Orbital Systems

TL;DR

This paper investigates the accuracy and limitations of low-order hybridization expansion impurity solvers like NCA and OCA in multi-orbital systems, revealing their dependence on the least correlated orbital.

Contribution

It analytically connects multi-orbital Green's functions to single-orbital cases and identifies the regimes where low-order methods are valid or break down.

Findings

Accuracy is governed by the least correlated orbital.

Spurious coupling suppresses correlation features like the Kondo resonance.

Numerical benchmarks identify parameter regimes of success and failure.

Abstract

Low-order hybridization expansion methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA) are widely used impurity solvers in the study of strongly correlated systems, yet their accuracy in genuine multi-orbital settings remains poorly understood. Using the decoupled orbital limit as a controlled reference point, we derive analytic results connecting multi-orbital restricted propagators and Green's functions to their single-orbital counterparts, identify the diagrammatic mechanisms responsible for the breakdown of low-order methods in multi-orbital settings, and determine their regimes of applicability. Our central finding is that the accuracy of these methods is governed by the least correlated orbital: i.e., the orbital with the most rapidly decaying retarded Green's function. That orbital's properties are transferred to all other orbitals through a spurious coupling generated by the truncated expansion, thereby suppressing correlation-induced features such as the Kondo resonance. This occurs even in orbitals that are themselves strongly correlated within single-orbital calculations using the same approximation scheme. We confirm this numerically across representative two-orbital model systems in the steady-state, systematically identifying the parameter regimes in which low-order methods succeed or fail. Our results provide a practical guide for assessing when insights from single-orbital calculations carry over to multi-orbital settings, and serve as a benchmark for the development and validation of higher-order multi-orbital impurity solvers.