Reevaluating Quantum Geometric Criteria for Itinerant Magnetic Instabilities

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.

• The paper demonstrates that quantum geometric measures from isolated susceptibility channels fail to predict magnetic instabilities in the presence of channel mixing.
• Using matrix-based GL and Hartree-Fock analyses, the study shows that full susceptibility and interaction tensors must be considered to diagnose FM and altermagnetic orders.
• Numerical examples reveal that even with dominant altermagnetic susceptibility, robust Hund's coupling can favor ferromagnetism, highlighting the complex interplay of geometry and interactions.

Reevaluating the Role of Quantum Geometry in Itinerant Magnetic Instabilities

Introduction

The connection between quantum geometric properties of Bloch states and correlated electron phenomena has prompted significant interest, especially regarding the diagnosis of magnetic instabilities in itinerant multiband systems. Prior investigations, notably in the context of ferromagnetism (FM) and altermagnetism (AM), have advanced the perspective that quantum geometric measures—particularly the quantum metric—can directly inform the presence and nature of incipient magnetic order. These proposals posit that the curvature and peak structure of single-channel (diagonal) susceptibilities, through their encoding of quantum geometric features, suffice as predictors of the ensuing magnetic phase. The present work systematically interrogates these claims by employing a rigorous, matrix-based analysis within the Ginzburg-Landau (GL) and Hartree-Fock (HF) frameworks, clarifying both the domain of validity and inherent limitations of geometric criteria for itinerant magnetism (2604.16782).

Formulation and Critique of Quantum Geometric Criteria

The central insight underlying geometric criteria is that the static bare magnetic susceptibility encodes the quantum distance between Bloch wavefunctions at distinct momenta, such that its curvature tensor incorporates a quantum geometric contribution. In systems with well-separated (decoupled) ordering channels, the corresponding curvature’s sign and magnitude, anchored in the quantum metric, influence the proximity to FM or AM instability. This relation, made explicit in multiband Hubbard models, has led to the assertion that geometric measures extracted from the bare FM or AM susceptibility alone may forecast magnetic phase transitions.

This work demonstrates, however, that such geometric diagnostics are neither universally sufficient nor necessary outside the narrow limit of complete channel decoupling. Using a matrix GL theory, it is shown that the onset of a magnetic instability is governed by the full interplay between the susceptibility tensor and the matrix of interaction strengths in the appropriate channel representation. The analysis reveals that consideration of only a single diagonal susceptibility, and its quantum geometric attributes, cannot generally resolve the prevailing instability if the interaction and susceptibility tensors possess off-diagonal couplings—i.e., if there is channel mixing. As a result, channel mixing content invalidates attempts to directly infer instability from the quantum geometry of an isolated channel.

Matrix-Based Criterion for Magnetic Instabilities

Developing the GL free energy to quadratic order in the order parameter, the work derives the instability criterion for multiorbital systems. A phase transition is signaled by the vanishing of the smallest eigenvalue of the stability matrix

$$\mathbb{M}(\mathbf{q}) = I - \chi^{(0)}(\mathbf{q}) U^{(s)},$$

where $\chi^{(0)}(\mathbf{q})$ is the full bare susceptibility tensor and $U^{(s)}$ encapsulates the spin-dependent interaction. Only in cases where both matrices are diagonal does the situation reduce to decoupled Stoner criteria for individual channels:

$$1 - [\chi^{(0)}(\mathbf{q})]^{aa} [U^{(s)}]_{aa} = 0$$

(no sum on $a$), thus validating the single-channel geometric approach. In more general settings, inter-channel coupling necessitates analyzing the entire matrix $\mathbb{M}(\mathbf{q})$.

The two-orbital models considered herein exemplify cases where symmetry can render both $\chi^{(0)}$ and $U^{(s)}$ diagonal, as in the presence of on-site Hubbard-Hund interaction. Even then, the prevailing instability depends on the competition of both the magnitude and shape (not solely the curvature) of diagonal susceptibilities and the respective interaction strengths.

Numerical and Analytical Illustration: FM vs. AM Order

By explicit computation of the susceptibility matrix for a representative two-orbital model, the study demonstrates that diagonal forms arise under significant symmetry constraints. In such scenarios, one can distinguish between FM and various AM channels ($\tau^x$, $\tau^y$, $\chi^{(0)}(\mathbf{q})$0). Critically, the analysis underscores that a larger bare susceptibility in an AM channel does not guarantee that AM is the leading instability; the strength of Hund's coupling, for example, can tip the balance toward FM, even in regions where AM susceptibilities outpace FM in the non-interacting limit. This contradicts prior interpretations that rely solely on the quantum geometric enhancement of single-channel susceptibilities as predictors of order, and demonstrates the essential role of the interplay between quantum geometry and microscopic interaction structure—consistent with results from RPA studies.

Implications and Outlook

The results have concrete implications for theoretical and computational modeling of complex multiorbital systems:

• Predictive Limitations of Quantum Geometry Alone: Reliance on quantum metric-derived criteria without explicit attention to channel structure and interactions can lead to incorrect predictions of phase behavior in itinerant electron systems, especially in cases with substantial channel entanglement.
• Channel Representation as a Diagnostic and Computational Tool: Reformulating the magnetic ordering problem in channel space, rather than the raw orbital basis, provides both physical transparency and computational simplification, especially when interaction matrices adopt block-diagonal or diagonal forms in this representation.
• Design of Materials and Spintronic Functionalities: For engineered materials targeting specific magnetic phases, the combined tuning of quantum geometry (via band structure engineering) and interaction parameters (especially Hund’s and interchannel exchange terms) is essential for robust order or dominant fluctuations.

Future research directions likely include the development of systematic criteria for channel decoupling in realistic materials, microscopic modeling incorporating longer-range or anisotropic interactions, and extensions of the analysis to spin-orbit coupled and noncollinear scenarios. Further, given the sensitivity of AM order to both band structure and interactions, experimental confirmation of quantum geometry-driven AM in candidate materials will require careful disentanglement of these factors.

Conclusion

This investigation establishes that, while quantum geometric quantities embedded in bare susceptibilities illuminate certain tendencies toward FM or AM order, they are insufficient as standalone predictors except under stringent decoupling conditions. Matrix-based stability analysis in the channel representation, capturing the full electron interaction and susceptibility structure, is indispensable for the accurate diagnosis of itinerant magnetic instabilities. The channel-based approach introduced herein thus provides a physically motivated, generalizable alternative to the more ad hoc single-channel geometric criteria, advancing the theoretical understanding of multiorbital magnetic phenomena and guiding future developments in the field.