Large-scale machine-learning-enabled simulations of the lightly hole-doped single-band double-exchange (ferromagnetic Kondo lattice) model showed that electronic phase separation is kinetically arrested: once antiferromagnetic correlations develop in the half-filled background, doped holes self-trap in the ferromagnetic spin clouds they create, hole evaporation is suppressed, and the late-time domain growth becomes sublogarithmic, L(t)∼(log⁡t)0.11L(t)\sim(\log t)^{0.11}, instead of the t1/3t^{1/3} law of Lifshitz–Slyozov–Wagner (LSW) coarsening theory. Those simulations treat the localized spins as classical vectors with adiabatic electron dynamics. The source work leaves open whether quantum effects — quantum spin dynamics, quantum tunneling of small magnetic polarons, and tunneling-induced evaporation of self-trapped holes — restore carrier mobility and standard LSW-type coarsening, or whether the arrest survives quantum fluctuations; it identifies quenched disorder, hole concentration, external fields, and Jahn–Teller/orbital degrees of freedom as contextual factors expected to influence carrier mobility in real materials, rather than as part of this open mechanism question.

Background

Colossal magnetoresistance (CMR) manganites — doped perovskite manganese oxides whose electrical resistance changes dramatically in a magnetic field — are believed to host electronic phase separation: nanoscale coexistence of hole-rich ferromagnetic metallic regions and hole-poor antiferromagnetic insulating regions. The minimal model for this physics is the single-band double-exchange model, also called the ferromagnetic Kondo lattice model: itinerant conduction electrons hop between lattice sites and are coupled by a strong on-site Hund exchange (the intra-atomic coupling that aligns the carrier spin with the localized spin) to one localized spin per site. At small hole doping away from half filling, the carriers gain kinetic energy by aligning the surrounding spins (the double-exchange mechanism), while the localized spins in the half-filled background experience an effective antiferromagnetic superexchange (a coupling between neighboring spins mediated by the electrons); the competition between these tendencies drives phase separation.

Simulating the real-time dynamics of such phase separation is difficult because the forces on the localized spins require solving the electronic problem for every spin configuration. Zhang and Chern (Phys. Rev. Lett. 127, 146401 (2021)) overcame this by training neural-network force fields on exact-diagonalization data (the exact numerical solution of the electron Hamiltonian, generated as 3500 spin-configuration snapshots from exact coupled spin–electron dynamics on a 30×30-site training lattice); the trained network predicts the electron-mediated exchange field for arbitrary spin configurations, enabling Landau–Lifshitz–Gilbert spin dynamics — the standard deterministic, dissipative classical equation of motion for magnetizations — on 100×100 lattices, within the adiabatic approximation in which the electrons stay in the instantaneous ground state of the current spin configuration. Precursor work had developed the coupled classical spin-electron dynamics and the machine-learning methodology itself.

The central finding was a kinetic arrest of phase separation. After a thermal quench, the average linear size L(t)L(t) of ferromagnetic clusters first follows the t1/3t^{1/3} law of Lifshitz–Slyozov–Wagner theory — the classical theory of coarsening for a conserved order parameter, in which larger domains of the minority phase grow by evaporation–condensation, absorbing carriers that diffuse out of smaller domains. Once antiferromagnetic correlations develop in the background, however, each doped hole becomes surrounded by a cloud of parallel spins that it itself creates through the double exchange; this cloud acts as a confining potential for the hole. Such self-trapping suppresses the evaporation on which LSW coarsening relies, and the late-time growth becomes sublogarithmic, L(t)∼(log⁡t)0.11L(t)\sim(\log t)^{0.11}. Holes that do escape become magnetic polarons — quasiparticles in which a handful of parallel localized spins (as few as five in this model) dress a single trapped charge — and these are essentially immobile in the classical dynamics, so Brownian-collision coarsening mechanisms also fail.

The simulation, however, is entirely classical: the localized spins are unit vectors obeying deterministic dynamics, and the carriers enter only adiabatically. Older literature argued that small magnetic polarons can move diffusively by quantum tunneling or by large paramagnetic fluctuations, although such tunneling is expected to be suppressed in CMR materials with large local spins, and no consistent microscopic treatment of tunneling-induced evaporation of holes from ferromagnetic clusters exists. In real materials, Jahn–Teller lattice distortions (coupling of the lattice to the degeneracy of the ege_g orbitals — the doubly degenerate subset of the manganese 3d3d orbitals in the perovskite crystal field) and orbital ordering accompany the polarons and are expected to stabilize them further, while quenched disorder can bind carriers into bound magnetic polarons and higher hole doping enhances carrier wavefunction overlap and tunneling. Whether quantum effects restore carrier mobility and standard coarsening, or whether the arrest survives quantum fluctuations, is unknown — this is the explicit question left open by the source work.

Problem Statement

Determine, by a controlled microscopic treatment that includes quantum carrier and spin dynamics, whether quantum effects overcome the correlation-induced self-confinement of doped holes in the phase-separating single-band double-exchange model, and thereby establish the asymptotic late-time coarsening law of the electronic phase separation — in particular whether the standard Lifshitz–Slyozov–Wagner growth of the characteristic domain size, L(t)∼t1/3L(t)\sim t^{1/3}, is restored, or whether the arrested sublogarithmic growth L(t)∼(log⁡t)0.11L(t)\sim(\log t)^{0.11} survives quantum fluctuations. A complete answer must (i) treat consistently, at the microscopic level, the tunneling-induced evaporation of holes out of ferromagnetic clusters and the quantum tunneling or fluctuation-assisted diffusion of small magnetic polarons through the antiferromagnetic background; and (ii) decide whether self-trapping persists when the localized spins and/or the carriers are quantized, given that quantum polaron tunneling is expected to be suppressed for the large spin magnitudes of CMR materials.

References

  1. P. Zhang and G.-W. Chern, "Arrested Phase Separation in Double-Exchange Models: Large-Scale Simulation Enabled by Machine Learning", Phys. Rev. Lett. 127, 146401 (2021). DOI: 10.1103/PhysRevLett.127.146401, https://doi.org/10.1103/PhysRevLett.127.146401; arXiv:2105.08221, https://arxiv.org/abs/2105.08221
  2. I. M. Lifshitz and V. V. Slyozov, "The kinetics of precipitation from supersaturated solid solutions", J. Phys. Chem. Solids 19, 35–50 (1961). DOI: 10.1016/0022-3697(61)90054-3, https://doi.org/10.1016/0022-3697(61)90054-3
  3. C. Wagner, "Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung)", Zeitschrift für Elektrochemie 65, 581–591 (1961). DOI: 10.1002/bbpc.19610650704, https://doi.org/10.1002/bbpc.19610650704
  4. J. Luo and G.-W. Chern, "Dynamics of electronically phase-separated states in the double exchange model", Phys. Rev. B 103, 115137 (2021). DOI: 10.1103/PhysRevB.103.115137, https://doi.org/10.1103/PhysRevB.103.115137; arXiv:1904.05252, https://arxiv.org/abs/1904.05252
  5. P. Zhang, P. Saha, and G.-W. Chern, "Machine learning dynamics of phase separation in correlated electron magnets", arXiv:2006.04205 (2020), https://arxiv.org/abs/2006.04205
  6. S. Zhang, P. Zhang, and G.-W. Chern, "Anomalous phase separation in a correlated electron system: Machine-learning–enabled large-scale kinetic Monte Carlo simulations", Proc. Natl. Acad. Sci. U.S.A. 119, e2119957119 (2022). DOI: 10.1073/pnas.2119957119; arXiv:2105.13304, https://arxiv.org/abs/2105.13304
  7. A. Tyberg, Y. Fan, and G.-W. Chern, "Machine learning force-field model for kinetic Monte Carlo simulations of itinerant Ising magnets", arXiv:2411.19780 (2024), https://arxiv.org/abs/2411.19780
  8. P. Zhang and G.-W. Chern, "Machine learning nonequilibrium electron forces for spin dynamics of itinerant magnets", npj Comput. Mater. 9, 32 (2023). DOI: 10.1038/s41524-023-00990-0, https://doi.org/10.1038/s41524-023-00990-0; arXiv:2112.12124, https://arxiv.org/abs/2112.12124
  9. S. Ghosh, S. Zhang, C. Cheng, and G.-W. Chern, "Kinetics of orbital ordering in cooperative Jahn-Teller models: Machine-learning enabled large-scale simulations", Phys. Rev. Materials 8, 123602 (2024). DOI: 10.1103/PhysRevMaterials.8.123602, https://doi.org/10.1103/PhysRevMaterials.8.123602; arXiv:2405.14776, https://arxiv.org/abs/2405.14776
  10. Y. Fan and G.-W. Chern, "Graph neural network force fields for adiabatic dynamics of lattice Hamiltonians", arXiv:2603.02039 (2026), https://arxiv.org/abs/2603.02039
  11. G.-W. Chern, Y. Fan, S. Zhang, and P. Zhang, "Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems", J. Magn. Magn. Mater. 642, 173898 (2026). DOI: 10.1016/j.jmmm.2026.173898, https://doi.org/10.1016/j.jmmm.2026.173898; arXiv:2604.11513, https://arxiv.org/abs/2604.11513
  12. G.-W. Chern, Y. Fan, S. Zhang, and P. Zhang, "Machine-learning force-field models for dynamical simulations of metallic magnets", AIP Adv. 16, 025031 (2026). DOI: 10.1063/9.0001056, https://doi.org/10.1063/9.0001056; arXiv:2602.18213, https://arxiv.org/abs/2602.18213
  13. T. Guthardt, M. Scheb, J. von Delft, F. Grusdt, and A. Bohrdt, "Finite-temperature real-time properties of magnetic polarons in two-dimensional quantum antiferromagnets", arXiv:2504.07715 (2025), https://arxiv.org/abs/2504.07715
  14. M. Callsen, J. H. Nyhegn, K. K. Nielsen, and G. M. Bruun, "Hole and spin dynamics in an anti-ferromagnet close to half filling", arXiv:2604.14039 (2026), https://arxiv.org/abs/2604.14039