Reevaluating Quantum Geometric Criteria for Itinerant Magnetic Instabilities

TL;DR

This paper critically re-examines quantum geometric criteria for itinerant magnetic instabilities, proposing a matrix-based approach that accounts for multiple magnetic channels, enhancing understanding and computational methods in multi-orbital systems.

Contribution

It introduces a rigorous matrix-based instability criterion in the channel representation, clarifying the role of quantum geometry in magnetic phase transitions.

Findings

Magnetic instabilities depend on the interplay between susceptibility tensor and spin interaction matrix.

Single-channel quantum geometric predictions are valid only under complete decoupling.

Channel representation improves transparency and computational efficiency.

Abstract

The interplay between quantum geometry and electron correlation has emerged as a compelling paradigm in quantum many-body physics. Recent studies have highlighted the diagnostic utility of quantum geometry in identifying magnetic instabilities within itinerant electron systems. In the present work, we critically re-examine these theoretical proposals. Using the Ginzburg-Landau framework within the Hartree-Fock mean-field approximation and accounting for multiple channels of magnetic ordering, we formulate a rigorous matrix-based instability criterion in the channel representation for generic two-orbital systems. Our results demonstrate that magnetic phase transitions are intricately governed by the interplay between the bare susceptibility tensor and the spin interaction matrix. Consequently, prior assertions that instabilities can be predicted solely from the quantum geometric structure of a single-channel susceptibility are valid only under complete channel decoupling in both the interaction and susceptibility matrices. By adopting the channel representation, our formulation achieves greater physical transparency and computational tractability compared to the conventional orbital-space approach, thereby furnishing a promising alternative for advancing theoretical studies of complex multi-orbital systems.