Researchers are increasingly investigating the complex behaviour of correlated materials with strong Hund’s coupling, but studies using cold atoms have largely concentrated on single-orbital Fermi-Hubbard models. Haoran Sun (Hefei National Research Center for Physical Sciences at the Microscale and University of Science and Technology of China), Erhai Zhao (George Mason University), and Youjin Deng (Hefei National Research Center for Physical Sciences at the Microscale and University of Science and Technology of China), alongside W. Vincent Liu et al, now demonstrate that fermionic atoms in the -bands of a hexagonal lattice provide a novel means to explore the interplay between Hund’s coupling and Mott physics. Their theoretical work predicts a surprising first-order transition between an itinerant ferromagnetic metal and a spin-1 antiferromagnetic insulator, driven by density, a finding that significantly advances our understanding of competing exchange mechanisms in multi-orbital systems and establishes -orbital Fermi gases as an ideal platform for future experimental investigation.
Fermionic Hubbard Model on a Hexagonal Lattice exhibits
This transition represents a significant advancement in understanding how competing exchange mechanisms drive magnetic order. The study unveils that ferromagnetism emerges at low fillings from a Flat band within the hexagonal lattice and persists to stronger interactions and higher fillings via a double-exchange mechanism. This mechanism aligns spins to minimise Hund-rule penalties, even at the expense of Dirac-fermion kinetic energy, establishing a robust ferromagnetic phase. The research establishes that long-range orbital order is suppressed by this geometric frustration, leading to a phase diagram dominated by the competition between distinct spin orders.
Furthermore, the team argues that the itinerant FM phase is remarkably robust, extending to higher fillings and stronger interactions, and is not solely dependent on the presence of a flat band. This robustness is attributed to the persistent influence of Hund’s coupling, which stabilises the ferromagnetic alignment. This breakthrough reveals a novel route to control magnetism in p-orbital systems, as the FM-AFM transition is solely governed by particle filling. The work opens exciting possibilities for quantum simulation, particularly in replicating and understanding the complex behaviour of correlated materials like cuprates, pnictides, and ruthenates. By successfully implementing the double-exchange mechanism, a cornerstone of magnetism in solids, in a quantum simulator, the research provides concrete theoretical predictions for current ultracold atom experiments and paves the way for exploring exotic magnetic phenomena in a highly controlled environment.
Fermion Interactions in Hexagonal Lattice Optical Lattices exhibit
Experiments were designed to harness ultracold-atom techniques for manipulating orbital states and tuning interactions within optical lattices. The study pioneered a method for projecting the kinetic term in the Hamiltonian onto axes parallel and perpendicular to the lattice bond directions, enabling precise control over hopping parameters. Specifically, the team defined operators annihilating fermions in orbitals oriented along bond directions by projecting Cartesian orbital operators onto unit vectors defining the longitudinal hopping term, e1,2 = ± √3/2x + 1/2 y, e3 = −y. This approach allows for the selective activation of hopping along specific lattice bonds, crucial for observing the predicted phase transitions.
To further refine the experimental setup, the researchers detailed the formulation of the neglected transverse hopping term, H⊥, which connects sites via orbitals oriented perpendicular to the bond direction. The team meticulously constructed the Hamiltonian, incorporating both longitudinal and transverse hopping terms to accurately represent the multi-orbital system. This precise control over orbital interactions is essential for realizing the predicted Hund metal regime and investigating its non-Fermi liquid properties. The innovative methodology enables the identification of distinct phases, from staggered order to double-exchange-driven ferromagnetism, using time-of-flight imaging, momentum-resolved spectroscopy, and in-situ microscopy. This combination of theoretical prediction and advanced experimental techniques lays the groundwork for incorporating quantum fluctuations using numerical methods such as DMFT or QMC, potentially revealing unconventional superfluidity and shedding light on the paramagnetic metal’s behaviour.
Density controls magnetism in hexagonal P-Band systems, revealing
Experiments predict that this transition is solely governed by particle filling, offering a new method for controlling magnetism in p-orbital systems. The team measured the emergence of ferromagnetism at low fillings originating from a flat band, which persists to stronger interactions and higher fillings through a double-exchange mechanism, where spins align to minimise Hund-rule penalties at the expense of Dirac-fermion kinetic energy. Results demonstrate that long-range orbital order is suppressed by geometric frustration on the hexagonal lattice, contrasting with observations on square lattices. The itinerant FM phase proves robust, extending to higher fillings and stronger interactions, and is stabilised by the double-exchange mechanism upon doping the AFM insulator.
Tests prove a first-order phase transition separates the FM and AFM phases, with itinerant carriers coexisting with localised moments throughout the FM phase. These localised states are associated with the flat band at low filling, while spin-1 moments form due to “Mottness” near half-filling. The study’s analysis yields that the persistence of ferromagnetism is largely due to Hund’s coupling, a cornerstone of magnetism in solids. Scientists recorded the band structure featuring two perfectly flat bands touching two dispersive bands with Dirac cones, as detailed in Figure 0.1. The interaction Hamiltonian, approximating the local optical potential, includes terms for on-site repulsion, inter-orbital repulsion, pair transfer, and Hund’s coupling, with the latter favouring spin alignment.
Hund’s coupling drives density-tuned magnetism in correlated materials
The emergence of ferromagnetism at low fillings, sustained by a double-exchange mechanism that prioritises Hund’s rule and tolerates kinetic energy costs for Dirac fermions, is a key finding. Acknowledging limitations, the authors note that their study provides a foundation for future investigations incorporating quantum fluctuations using advanced numerical techniques like Dynamical Mean-Field Theory (DMFT) or Quantum Monte Carlo (QMC). They also suggest that further research could illuminate the non-Fermi liquid properties potentially present in the paramagnetic metal near the FM or AFM transitions, and explore the possibility of unconventional superfluidity. These findings are significant as they expand our understanding of correlated electron systems and could guide the development of novel quantum materials with tailored magnetic and electronic properties.