Double-exchange ferromagnetism of fermionic atoms in a $p$-orbital hexagonal lattice

Abstract

A large class of correlated quantum materials feature strong Hund's coupling. Yet cold-atom quantum simulators have so far focused primarily on single-orbital Fermi-Hubbard systems near a Mott insulator. Here we show that repulsively interacting fermions loaded into the $p$-bands of a hexagonal lattice offer a unique platform to study the interplay of "Hundness" and "Mottness." Our theory predicts that the orbital degrees of freedom, despite geometric frustration, produce a rich phase diagram featuring a competing itinerant ferromagnetic (FM) metal and a spin-1 antiferromagnetic (AFM) insulator, with a surprising first-order transition between them controlled by density near half-filling. Ferromagnetism emerges at low fillings from the flat band and persists to stronger interactions and higher fillings via a double-exchange mechanism, where spins align to avoid Hund-rule penalties at the expense of Dirac-fermion kinetic energy. We further argue that the paramagnetic regime is a correlated "Hund metal." $p$-orbital Fermi gases thus provide an ideal experimental setting to investigate competing exchange mechanisms in multi-orbital systems with coexisting localized and itinerant spins.

Figures (4)

• Figure 1: (a) Schematic of the hexagonal lattice, showing the A and B sublattices and the nearest-neighbor vectors. Transverse hopping $t_perp$ is neglected. (b) The non-interacting band structure for $t_\perp = 0$, featuring two inequivalent Dirac cones and two perfectly flat bands at energies $\epsilon = \pm 3/2t$.
• Figure 2: Overview of spin order and phase competition. (a) Mean-field phase diagram in the $\mu$-$U$ plane (units: $t\equiv 1$). The color map shows the magnitude of the order parameter for the ferromagnetic (FM) phase (total magnetization $M$, red) and the antiferromagnetic (AFM) phase (staggered magnetization $m_s$, blue). The system exhibits a paramagnetic (PM) phase in the white region where both order parameters are zero. (b) A detailed slice at strong interaction $U=10$, showing the order parameters (staggered magnetization $m_s$, blue line; total magnetization $M$, red line) and the total filling $\langle n \rangle$ (black dashed line) as a function of chemical potential $\mu$. The discontinuous jumps in $m_s$, $M$, and $\langle n \rangle$ signal a first-order phase transition between the FM and AFM states. (c) A slice at half-filling ($\langle n \rangle = 2$). The plot shows the staggered magnetization $m_s$ (solid blue line) and the corresponding energy gap (dashed orange line) as a function of interaction strength $U$. The system undergoes a continuous, second-order phase transition into the AFM state at a critical interaction of $U_c \approx 3.253t$, where the energy gap opens.
• Figure 3: Schematics of competing mechanisms governing the system's ground state. (a) Geometric orbital frustration on the hexagonal lattice. Unlike in square lattice where a simple ferro-orbital alignment is unfrustrated (left), the hexagonal geometry makes it impossible to simultaneously minimize the kinetic energy across all bonds due to $p$-orbital geometry (right). This conflict strongly suppresses the tendency towards long-range orbital order. (b) The double-exchange mechanism. (i) A FM background allows holes to delocalize freely. (ii) In AFM background, the hopping process is suppressed. (iii) Hopping is energetically penalized by a large on-site Hund's coupling, $J_H=U/4$ (blurred spins). This blockade of hopping in the AFM state makes the FM state energetically favored at finite doping.
• Figure A1: The orbital stiffness kernel $K_\theta(0,0;\phi_z)$ as a function of the staggered magnetization field amplitude $\phi_z$. The kernel remains positive definite across the entire range from the semi-metallic phase ($\phi_z=0$) to the gapped AFM phase ($\phi_z > 0$). A positive value corresponds to a finite mass for the orbital fluctuations, indicating that the orbital excitations are gapped and the system is stable against orbital ordering.