Kinetic Energy Driven Ferromagnetic Insulator

Abstract

We construct a minimal model of interacting fermions establishing a ferromagnetic insulating phase. It is based on the Hubbard model on a trimerized triangular lattice in the regime of $t\gg |t^\prime|>0$ with $t$ and $t^\prime$ the intra- and inter-trimer hopping amplitudes, respectively. At the $\frac{1}{3}$-filling, each trimer becomes a triplet spin-1 moment, and the inter-trimer superexchange is ferromagnetic with $J =- \frac{2}{27}\frac{t^{\prime 2}}{t}$ in the limit of $U/t=+\infty$. As $U/t$ becomes finite, the antiferromagnetic superexchange competes with the ferromagnetic one. The system enters into a frustrated antiferromagnetic insulator when $λ>U/t\gg 1$ where $λ$ is a constant at the order of 10. In contrast, a similar analysis to the trimerized Kagome lattice shows that only the antiferromagnetic superexchange exists at $\frac{1}{3}$-filling.

Figures (4)

• Figure 1: (a) The trimerized triangular lattice with hopping strengths — the intra-trimer hopping $t$ and the inter-trimer hopping $t^\prime$ represented by the solid and dashed lines, respectively. Each trimer is filled with 2 electrons. (b) The bases of the sector with $S_{\text{tot}}=0$ are generated by the hole's hopping around the trimer, sequentially denoted as $|1\rangle=c_{1\uparrow}^\dagger c_{2\downarrow}^\dagger |\Omega\rangle, |2\rangle=c_{3\downarrow}^\dagger c_{1\uparrow}^\dagger |\Omega\rangle, |3\rangle=c_{2\uparrow}^\dagger c_{3\downarrow}^\dagger |\Omega\rangle, |4\rangle=c_{1\downarrow}^\dagger c_{2\uparrow}^\dagger |\Omega\rangle, |5\rangle=c_{3\uparrow}^\dagger c_{1\downarrow}^\dagger |\Omega\rangle$, and $|6\rangle=c_{2\downarrow}^\dagger c_{3\uparrow}^\dagger |\Omega\rangle$.
• Figure 2: Particle-hole excitations leading to (a) the FM superexchange at $U/t >\lambda$ and (b) the AFM one at $\lambda > U/t \gg 1$, respectively, with the transition value $\lambda\sim 10$. For the FM case, the excitation energy $E_{ex}\sim t$ with only single occupations analogues to charge-transfer insulators, while $E_{ex}\sim U$ for the AFM case manifesting the Mott physics characterized by the double occupancy.
• Figure 3: (a) The phase diagram at the $\frac{1}{3}$-filling for $0 < t^\prime/t \le 0.6$. DMRG calculations are applied for the system on a tilted cylinder with the size of $L_x \times L_y \times 3$ to determine the boundary between the fully spin-polarized FM insulating state and the unpolarized AFM one, which is marked by a solid line. The AFM-FM transition is determined based on $L_x=6, L_y=4$. The data of $L_x=\infty$ and $L_y=3$ are also calculated yielding similar results. The decay length $\xi_e$ is extracted from the single-electron Green's function by extending the system to the quasi-1D limit of $L_x \to \infty$ (see SM. Sect. G). Charge localizations are found down to certain values of $U$, and the data with $\xi_e$ less than the inter-trimer distance $\sqrt 3 a$, are marked by blue dots for reference. The metal-insulator transition boundary (black dashed line) is estimated from the relation $U_c / W \approx \sqrt{3}/2$, where $W$ is the free bandwidth. (b) The relative spontaneous magnetization $M_s / M_{\text{s},\max}$, with $M_s = \sqrt{\langle \boldsymbol{S}_{\text{tot}}^2 \rangle}$ and $M_{\text{s},\max} = \sqrt{S_{\max} (S_{\max} + 1)}$ where $S_{\max} = L_x\times L_y$ equals to the number of trimers. It reaches 1 at $U/t \approx 14$, confirming full polarization. (c) The single-electron gap, defined as $\Delta = E_{N=49}^{\text{GS}} + E_{N=47}^{\text{GS}} - 2E_{N=48}^{\text{GS}}$, exhibits non-zero values except at $U=0$. The $U=0$ result has exact zero gap because there are degenerate single-particle states at Fermi energy. The red solid line is the band gap when electrons are fully spin polarized for comparison. The deviation in FM region is due to the fact that spins remain fully polarized for doping a hole but partially flip when adding an electron.
• Figure 4: (a) The trimerized Kagome lattice in which trimers form a triangular lattice. (b) The Kagome trimer lattice. Two neighboring trimers are connected via one bond in (a) and two bonds in (b), respectively.