Magnetic fluctuations driven by quantum geometry

Using quantum distance, magnetic susceptibility in the non-interacting limit can be rigorously split into two contributions: one arising solely from band dispersion, while the other stems from quantum geometric contributions. This work applies this decomposition to LaFeAsO and Pb9Cu(PO4)6O to demonstrate that their dominant magnetic fluctuations originate from the geometric contribution, highlighting its essential role in governing magnetic fluctuations in multi-band systems.

The gist

The dominant magnetic fluctuations in both LaFeAsO and Pb9Cu(PO4)6O are induced by the contribution referred to as the geometric term, which is determined by quantum geometry.

Theoretical Framework for Decomposition

The irreducible static susceptibility, defined as the magnetic susceptibility in the non-interacting limit, is decomposed into a band term and a geometric term:

χ0 (q) = χ0 band(q) + χ0 geom(q). (2)

The band contribution is given by:

χ0 band(q) = X k X λλ' δλλ'Fλλ'(k, q), (4)

where Fλλ'(k, q) is the Lindhard function defined as:

(5)

with d2λλ'(k, k') representing the quantum distance between Bloch states.

The geometric contribution is explicitly given by:

χ0 geom(q) = X k X λλ' [1 − d2λλ'(k, k + q) − δλλ'Fλλ'(k, q)], (3)

In systems with trivial band geometry—where the overlap of Bloch states is negligible, i.e., ⟨uλ(k)uλ′(k′)⟩ = δλλ'—the quantum distance simplifies to d2λλ'(k, k') = 1−δλλ', causing the geometric contribution to vanish.

Application to LaFeAsO: Antiferromagnetic Fluctuations

Starting from first-principles calculations in density functional theory (DFT), a tight-binding model was constructed for LaFeAsO. The analysis of the static irreducible susceptibility χ0(q) revealed that:

The band contribution, χ0 band(q), shows only broad features and does not favor the ordering at Q.

In stark contrast, the geometric contribution, χ0 geom(q), exhibits dominant peaks exactly at these antiferromagnetic wave vectors. Therefore, it is concluded that the stripe-type antiferromagnetic instability in LaFeAsO is decisively driven by the quantum geometry of the Bloch states.

Application to Pb9Cu(PO4)6O: Ferromagnetic Fluctuations

For Pb9Cu(PO4)6O, calculations were performed considering only the spin of Cu 3d orbitals. The total static susceptibility χ0(q) exhibited a peak at QΓ = (0, 0, 0), indicating a ferromagnetic fluctuation. Upon decomposition:

The band term shows peaks at QH = (1/2, 1/2, 1/2), which would favor antiferromagnetic fluctuations if the geometric term were absent.

However, the geometric contribution, χ0 geom(q), is negative for all q, with particularly large magnitude at QH, thereby suppressing the antiferromagnetic tendency. Consequently, the ferromagnetic fluctuation at QΓ as the leading instability is stabilized by quantum geometry.

Persistence in Interacting Systems

The study employed the random phase approximation (RPA) to calculate spin susceptibility in interacting systems to address whether this geometric origin persists. The results confirm that:

In LaFeAsO, the peak at Q = (1/2, 0) is enhanced by interactions, consistent with previous studies.

In Pb9Cu(PO4)6O, the ferromagnetic peak at Q = (0, 0, 0) is likewise intensified, in agreement with Ref. [46]."

These findings indicate that the geometric origin of magnetic fluctuations persists at least in the weak-coupling regime.

Conclusion and Significance

The research demonstrates that quantum geometry plays a decisive role in driving both antiferromagnetic and ferromagnetic fluctuations across different materials. The band–geometric decomposition provides a unified and quantitative framework to organize such effects, suggesting it can be broadly extended to other correlated electron systems to systematically classify magnetic fluctuations by disentangling band-dispersion and wavefunction-geometry contributions. The geometric term is indispensable for stabilizing the ferromagnetic response in Pb9Cu(PO4)6O, showing that dispersion alone is insufficient. Furthermore, this framework may elucidate how wavefunction geometry governs the competition between distinct magnetic instabilities and the resulting superconducting gap structures in iron-based superconductors.

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Here are the specific improvements that could be made to AI systems, based on the findings of this scientific paper:

1. Improve Materials Discovery and Screening for Magnetic Properties:

2. Develop Predictive Models for Multi-Orbital Magnetic Interactions:

3. Enhance Understanding of Quantum Geometric Effects in Strongly Correlated Systems:

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1. Improve Materials Discovery and Screening for Magnetic Properties:

AI systems could be improved to accurately predict the dominant magnetic ground state (ferromagnetic vs. antiferromagnetic) in novel multi-band materials by incorporating a geometric contribution term derived from quantum geometry (Hilbert-Schmidt distance). This moves beyond simple band dispersion or Fermi surface nesting alone.

The improved system could perform:

• Identify new classes of magnetic materials where magnetism is governed by orbital overlap and wavefunction geometry, rather than just energy differences.

• Screen vast chemical spaces for compounds likely to exhibit specific magnetic orders (e.g., stripe-type antiferromagnetism in LaFeAsO analogues) by calculating the sign and magnitude of the geometric susceptibility term, which dictates whether a material favors AFM or FM fluctuations at a given wave vector Q.

2. Develop Predictive Models for Multi-Orbital Magnetic Interactions:

AI systems could be refined to treat irreducible susceptibilities as a sum of two distinct terms: one purely dependent on band dispersion (Fermi surface nesting) and the other dependent on wavefunction overlap (quantum geometry).

The improved system could perform:

• Decompose complex magnetic response calculations into physically interpretable components. When analyzing experimental data or first-principles simulations, the AI would explicitly quantify how much of the observed magnetic instability is due to band effects versus wavefunction geometry effects.

• Develop robust models for multi-orbital systems where simple nesting criteria fail; the system would correctly prioritize orbital overlap effects when predicting spin fluctuation tendencies.

3. Enhance Understanding of Quantum Geometric Effects in Strongly Correlated Systems:

AI systems could be trained on the formalisms presented (like the band-geometric decomposition and the use of quantum distance) to systematically analyze and interpret results from quantum chemistry or condensed matter simulations involving complex electronic structures.

The improved system could perform:

• Analyze first-principles calculations (DFT) for complex materials, automatically extracting and calculating the geometric term to determine if it is driving observed magnetic instabilities (as demonstrated in LaFeAsO).

• Provide a quantitative framework for understanding how quantum geometry stabilizes or suppresses competing magnetic orders (e.g., suppressing AFM fluctuations to stabilize FM fluctuations in Pb9Cu(PO4)6O).

• Systematically classify magnetic instabilities across the family of iron-based superconductors by applying this band–geometric decomposition, potentially linking specific fluctuation types (like stripe-type vs. Q=(0,0,0) ferromagnetic) directly to the underlying geometry of the Bloch states.