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arXiv:2606.11302

Ferromagnetism from the geometry of localized wavefunctions in moiré systems

Miguel Gonçalves, Sarang Gopalakrishnan

Published Jun 9, 2026• 0 citations
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Abstract

We present a mechanism for ferromagnetism in narrow bands consisting of Anderson-localized states. We exploit single-particle localization to derive a controlled theory of exchange interactions within the narrow band. For quasiperiodic systems with a half-filled moiré band, we show that the critical interaction strength for ferromagnetism is highly sensitive to the geometry of real-space overlaps between localized orbitals: we find well-defined resonances at which ferromagnetism sets in for interaction energies that are far lower than the gap to other bands. Near these resonances, all the approximations in our theory are controlled, so our critical point predictions are quantitative. We show examples both in one and two dimensions. Our work identifies a route to ferromagnetism based on the geometry of real-space wavefunctions, distinct from previously found mechanisms based on the quantum geometry of Bloch bands.

DOI
10.48550/arxiv.2606.11302

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We present a mechanism for ferromagnetism in narrow bands consisting of Anderson-localized states. We exploit single-particle localization to derive a controlled theory of exchange interactions within the narrow band. For quasiperiodic systems with a half-filled moiré band, we show that the critical interaction strength for ferromagnetism is highly sensitive to the geometry of real-space overlaps between localized orbitals: we find well-defined resonances at which ferromagnetism sets in for interaction energies that are far lower than the gap to other bands. Near these resonances, all the approximations in our theory are controlled, so our critical point predictions are quantitative. We show examples both in one and two dimensions. Our work identifies a route to ferromagnetism based on the geometry of real-space wavefunctions, distinct from previously found mechanisms based on the quantum geometry of Bloch bands.
Quantum geometry quantifies how the single-particle Bloch wavefunction changes in phase and amplitude across the Brillouin Zone. In multi-orbital systems where bands have strongly mixed orbital composition, quantum geometry plays a vital role in determining the ground state and low-energy properties of interacting electronic systems. In this work, we show that Mott metal-insulator transitions, as well as transitions between different magnetic orders within the Mott insulating phase, can be driven by the quantum geometry of the underlying Bloch band, thereby providing a mechanism complementary to conventional bandwidth-tuned Mott transitions. By studying the Kane-Mele-Hubbard model using exact diagonalization, we demonstrate that in in half-filled and topologically-trivial bands, quantum geometric properties of the Bloch states alone can act as a tuning knob for Mott metal-to-insulator and affect the competition between ferromagnetism and antiferromagnetism. We show that both transitions may be heuristically understood via non-local Coulomb scattering in a basis of exponentially localized Wannier functions. These results highlight the role of quantum geometry beyond topological settings as a governing principle for conventional Mott and magnetic physics in multi-orbital and moiré materials.
We demonstrate that a mean field approximation can be confidently employed in quasiperiodic moiré systems to treat interactions and quasiperiodicity on equal footing. We obtain the mean field phase diagram for an illustrative one-dimensional moiré system that exhibits narrow bands and a regime with non-interacting multifractal critical states. By systematically comparing our findings with existing exact results, we identify the regimes where the mean field approximation provides an accurate description. Interestingly, in the critical regime, we obtain a quasifractal charge density wave, consistent with the exact results. To complement this study, we employ a real-space implementation of the time-dependent Hartree-Fock, enabling the computation of the excitation spectrum and response functions at the RPA level. These findings indicate that a mean field approximation to treat systems hosting multifractal critical states, as found in two-dimensional quasiperiodic moiré systems, is an appropriate methodology.
We study the interplay between electronic interactions and quasiperiodicity in a one-dimensional narrow-band system, focusing on ground-state and low-energy excitation properties. Using band projection as low-energy effective approach, we show that a projection restricted to first order in the interaction strength fails to reproduce the correlated phase diagram. This contrasts with the standard success of first-order band projection in translationally invariant flatband systems and highlights the essential role of virtual processes involving remote bands in quasiperiodic settings. By incorporating second-order interband contributions perturbatively, we obtain an effective Hamiltonian that quantitatively reproduces the exact phase iagram previously obtained using density matrix renormalization group calculations, including the transition between a Luttinger liquid and a charge-density-wave phase and the crossover to a quasifractal charge-density-wave regime at strong quasiperiodicity. We further use this controlled framework to investigate low-energy neutral excitations and the optical conductivity, identifying clear dynamical signatures distinguishing the different phases. Our results establish second-order band projection as a reliable tool for correlated quasiperiodic narrow-band systems and suggest a promising route for studying interacting quasiperiodic and moiré materials beyond one dimension.
Recent experiments in moiré quantum materials exhibit quantized Hall states without an external magnetic field, motivating continuum mechanisms based on smooth moiré-periodic pseudospin textures. We present a controlled theory of skyrmion Chern bands generated by such textures. An exact local SU(2)SU(2)SU(2) transformation reveals an emergent non-Abelian gauge field; for large branch splitting we perform an operator-level Schrieffer-Wolff expansion, yielding a single-branch Hamiltonian together with systematically dressed physical operators that define the projected interacting theory beyond strict adiabaticity. The leading dynamics is governed by a U(1)U(1)U(1) Berry connection whose flux is set by the skyrmion density, while controlled non-adiabatic corrections are fixed by the texture's real-space quantum geometric tensor. In a Landau-level representation built from the averaged emergent field, moiré-periodic modulations induce Umklapp-resolved deformations of Girvin-MacDonald-Platzman kinematics and microscopic sources of excess optical quantum weight above the topological lower bound. Assuming a gapped Hall phase, we further derive a skyrmion-crystal effective field theory with a universal Berry-phase term and a noncommutative magnetophonon. Our results provide experimentally accessible signatures for twisted transition-metal dichalcogenide homobilayers and rhombohedral graphene aligned with hexagonal boron nitride.
Despite its abundance in nature, predicting the occurrence of ferromagnetism in the ground state is possible only under very limited conditions such as in a flat band system with repulsive interaction or in a band with a single hole under infinitely large Coulomb repulsion, etc. Here, we propose a general condition to achieve saturated ferromagnetism based on the quantum geometry of electronic wave functions in itinerant electron systems. By analyzing the spin excitations of multi-band repulsive Hubbard models with an integer band filling, relevant to either ferromagnetic insulators or semimetals, we show that quantum geometry stabilizes the Goldstone mode in the strongly correlated limit. Our theory indicates the stability of ferromagnetism in a large class of insulators and semimetals other than the previously studied flat band systems and their variants. Moreover, we rigorously prove that saturated ferromagnetism is forbidden in any system with trivial quantum geometry, which includes every half-filled system. We believe that our findings reveal a profound connection between quantum geometry and ferromagnetism, which can be extended to various symmetry-broken ground states in itinerant electronic systems.
In non-interacting systems, bands from non-trivial topology emerge strictly at half-filling and exhibit either the quantum anomalous Hall or spin Hall effects. Here we show using determinantal quantum Monte Carlo and an exactly solvable strongly interacting model that these topological states now shift to quarter filling. A topological Mott insulator is the underlying cause. The peak in the spin susceptibility is consistent with a possible ferromagnetic state at T = 0. The onset of such magnetism would convert the quantum spin Hall to a quantum anomalous Hall effect. While such a symmetry-broken phase typically is accompanied by a gap, we find that the interaction strength must exceed a critical value for this to occur. Hence, we predict that topology can obtain in a gapless phase but only in the presence of interactions in dispersive bands. These results explain the recent quarter-filled quantum anomalous Hall effects seen in moiré systems.