Quantum-geometry-driven exact ferromagnetic ground state in a nearly flat band

Abstract

We construct a Hubbard model with a nearly flat band whose quantum geometry can be tuned independently of the energy dispersion and the Coulomb interaction. We show that, when the nearly flat band is half-filled, the exact ground state of the model exhibits ferromagnetism and that this ferromagnetism is stabilized by the quantum metric through the spin stiffness. Furthermore, we demonstrate that tuning the quantum geometry alone drives a magnetic phase transition. Our nonperturbative results without resorting to mean-field approximations reveal the quantum-geometric origin of ferromagnetism and the underlying many-body physics in dispersive-band systems.

Figures (7)

• Figure 1: (a) Tasaki's delta chain and 1D periodic chains with nearest neighbor hoppings. The lattice vector is represented by the navy-blue arrow. (b) Energy dispersion of $\hat{H}_{0}$ for $(s,t,u,v,\lambda) = (1/20,1/2,1/2,1,\sqrt{2})$. The middle bands are $(N_{\rm sub}-2)$-fold degenerate. (c) The unitary transformation from $\hat{H}_{0:\rm T} + \hat{H}_{0:\rm NN}$ to $\hat{H}_{0}$ by $\mathcal{U}_\theta$ is schematically illustrated.
• Figure 2: (a) The orange region denotes where the ground state is saturated ferromagnetism, while blue and purple regions denote where the saturated ferromagnetic state is unstable. In the blue region, $\mathcal{D}_{\rm spin}<0$ is additionally satisfied. (b) Magnon dispersion for $U = 1.36$ and several values of $\theta$. The orange and blue colors correspond to $\theta = 0$ and $\theta = \pi /4$, respectively.
• Figure 3: $\theta$-dependence of the spin stiffness for $U = 1.36$. Panel (a) shows $\mathcal{D}_{\rm geom}$ (blue squares), $\mathcal{D}_{\rm hexc}$ (orange circles), $\mathcal{D}_{\rm spin }$ (green triangles), while panel (b) shows $\mathcal{D}_{\rm met}$ (dark-blue squares), $\mathcal{D}_{\rm \rm gcov}$ (dark-magenta circles), $\mathcal{D}_{\rm con }$ (dark-cyan triangles). In (a), the region highlighted by the red oval indicates the negative $\mathcal{D}_{\rm spin}$.
• Figure 4: (a) $\bm q$ dependence of $\bar{\chi}(\bm q)$ for several values of $\theta$. The blue and orange lines correspond to $\theta = 0$ and $\pi/4$, respectively.
• Figure 5: Schematic illustration of $\hat{\alpha}^\dagger_{{\rm o},\sigma},\hat{\alpha}^\dagger_{{\rm r},\sigma}$, and $\hat{\alpha}^\dagger_{{\rm l},\sigma}$.