Quantum geometry, localization, and topological bounds of spin fluctuations

The gist: Crystalline dislocations are shown to strongly enhance the quantum metric, establishing a direct link between lattice topology and Hilbert-space geometry, which controls magnon localization and topological phases.

Quantum Geometry in Condensed Matter

Quantum geometry is central to understanding fundamental phenomena in condensed matter physics In parameter space, quantum geometry is fully characterized by the quantum geometric tensor (QGT), also known as the Fubini–Study metric The imaginary part of the QGT corresponds to the well-studied Berry curvature, which plays a central role in understanding adiabatic phase evolution and transport phenomena such as the anomalous Hall effect In contrast, much less attention has been given to the real component of the QGT—the quantum metric—which quantifies the local distance between nearby quantum states in the Hilbert space Quantum metric is crucial in diverse material properties and phenomena, bridging geometric and topological aspects of quantum matter It contributes to intrinsic nonlinear conductivities in noncentrosymmetric systems and determines the stability of fractional Chern insulators.

Magnonic Excitations and Topological States

Magnetic order supports magnonic excitations—the bosonic counterparts of spin waves—that can host topological states in the presence of magnetic textures or spin–orbit coupling In this regime, the corresponding Bogoliubov–de Gennes Hamiltonian exhibits magnon bands with nonzero Chern numbers, i.e., momentum-space integrals of the Berry curvature, leading to chiral edge states When magnons interact with crystalline defects such as disorder or lattice dislocations, new tools are required to characterize the real-space topology Crystal dislocations generate local strain fields that break translational symmetry, modifying the magnon band structure and producing localized defect states When parity symmetry is broken, magnonic states bound to the dislocation can become Z2 topologically protected.

Dislocation-Induced Geometric Enhancement

In this Letter, it is demonstrated that crystalline dislocations lead to a pronounced enhancement of the quantum metric, revealing a fundamental link between lattice topology and quantum geometry The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure This enhancement tracks the localization of magnonic states, which become increasingly delocalized in regions of large quantum metric Crystalline defects thus act as geometric degrees of freedom that tune the quantum metric, control magnon localization and topological phases.

Ordered Arrays of Dislocations

In an ordered array of dislocations, the bulk magnon spectrum is shown to exhibit a gap at the Γ point—signaling the absence of The two lowest-energy bands are remarkably flat, reflecting non-propagating modes with vanishing group velocity Such flatness suppresses kinetic energy, thereby enhancing interaction effects and favoring the emergence of correlated phases. These states correspond to modes localized at the dislocations. The ordered array of dislocations give rise to strain-induced modifications of the band geometry, yielding a measurable enhancement of the quantum metric tensor near defect cores, signaling increased Bloch-state overlap and reduced magnonic localization length.

Disordered Arrays and Hexatic Phase

In the disordered limit, at high defect densities corresponding to the hexatic phase, magnonic states become topologically trivial and strongly localized The transition to a topologically trivial phase (B = 0) coincides with a pronounced enhancement of the quantum metric, indicating the formation of highly localized magnonic states Beyond this point, coincident with the onset of the hexatic phase, the system becomes dominated by localized states. This geometric response thus provides a microscopic link between defect structure, quantum-state geometry, and the emergence of topological magnon channels in nontrivial phases.

Real-Space Definition of Quantum Metric

The quantum metric is generalized using a real-space approach when translational symmetry is lacking, where the quantum metric density g(r) is given by g(r) = -1/2π([Xµ, P][Xµ, P]) r,r The local quantum metric g(r), computed for various dislocation concentrations, is displayed in panels (a) and (b) The behavior of G is examined alongside the stability of the topological magnonic phase, quantified by the Bott index B, a robust real-space invariant of bulk topology As the dislocation concentration increases, lattice distortions progressively alter the band structure and, beyond a critical threshold ρ ∼ 0.5 − 0.6, induce partial gap closure. This transition is accompanied by a sharp enhancement of the quantum metric, signaling the appearance of strongly localized in-gap modes bound to dislocation cores. The concomitant evolution of G and B establishes a direct correspondence between disorder-induced geometric amplification and the breakdown of band topology.

Conclusion

Crystalline defects are identified as active geometric degrees of freedom, offering a tunable route to control quantum geometry and emergent topology Our results identify crystalline defects as active geometric degrees of freedom, offering a tunable route to control quantum geometry and emergent topology While the quantum geometric tensor has recently been accessed by spin- and angle-resolved photoemission spectroscopy (ARPES) in electronic systems, its detection in magnonic platforms remains challenging Recent progress in nonlinear thermal and charge transport, together with magnetic circular dichroism (MCD) techniques, provide a viable experimental route for accessing quantum geometry in these systems. The paper concludes that topological crystalline defects strongly enhance the quantum metric, establishing a direct link between lattice topology and Hilbert-space geometry. The study demonstrates that dislocations act as geometric degrees of freedom that tune the quantum metric, control magnon localization and topological phases.

How it works

1. Crystalline dislocations generate local strain fields that break translational symmetry, modifying the magnon band structure and producing localized defect states

2. The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure

3. In ordered arrays of dislocations, the bulk magnon spectrum exhibits a gap at the Γ point—signaling the absence of

4. In disordered arrays, dislocation-driven geometry expands the accessible topological phase space and enables transitions to disorder-induced topological phases

5. The real-space quantum metric density g(r) is given by g(r) = -1/2π([Xµ, P][Xµ, P]) r,r This real-space expression represents the real-space bosonic analogue of the quantum metric density.

6. The transition to a topologically trivial phase (B = 0) coincides with a pronounced enhancement of the quantum metric, indicating the formation of highly localized magnonic states.

7. Beyond a critical threshold ρ ∼ 0.5 − 0.6, lattice distortions induce partial gap closure, which is accompanied by a sharp enhancement of the quantum metric.

Key Findings Enumerated

**- Crystalline dislocations strongly enhance the quantum metric, establishing a direct link between lattice topology and Hilbert-space geometry The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure. This enhancement tracks the localization of magnonic states, which become increasingly delocalized in regions of large quantum metric. Crystalline defects thus act as geometric degrees of freedom that tune the quantum metric, control magnon localization and topological phases. The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure. This enhancement tracks the localization of magnonic states, which become increasingly delocalized in regions of large quantum metric. Crystalline defects thus act as geometric degrees of freedom that tune the quantum metric, control magnon localization and topological phases. The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure. This enhancement tracks the localization of magnonic states, which become increasingly delocalized in regions of large quantum metric. Crystalline defects thus act as geometric degrees of freedom that tune the quantum metric, control magnon localization and topological phases. The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure. This enhancement tracks the localization of magnonic states, which become increasingly delocalized in regions of large quantum metric. Crystalline defects thus act as geometric degrees of freedom that tune the quantum metric, control magnon localization and topological phases. The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure. This enhancement tracks the localization of magnonic states, which become increasingly delocalized in regions of large quantum metric. Crystalline defects thus act as geometric degrees of freedom that tune the quantum metric, control magnon localization and topological phases. The presence of dislocations reshapes the local Hilbert-space geometry, amplifying geometric responses associated with the underlying magnonic band structure.

1. Tunable Quantum Geometry for Material Design: The ability to tune quantum geometry through crystalline defects allows AI systems to design novel materials by controlling magnon localization and topological phases, specifically by identifying how defect-induced geometric enhancement controls their localization and topological protection.

2. Disorder-Driven Topological Phase Transitions: AI can leverage the behavior of disordered arrays where dislocation-driven geometry expands the accessible topological phase space and enables transitions to disorder-induced topological phases, enabling the design of materials that transition between trivial and nontrivial states based on controlled defect concentrations.

3. Real-Space Geometric State Characterization: By utilizing real-space approaches like Eq. (4), AI can compute the real-space bosonic analogue of the quantum metric density to map out local geometric properties, providing a microscopic link between defect structure, quantum-state geometry, and the emergence of topological magnon channels.