Quantum-geometric dipole: a topological boost to flavor ferromagnetism in flat bands

Abstract

Robust flavor-polarized phases are a striking hallmark of many flat-band moiré materials. In this work, we trace the origin of this spontaneous polarization to a lesser-known quantum-geometric quantity: the quantum-geometric dipole. Analogous to how the quantum metric governs the spatial spread of wavepackets, we show that the quantum-geometric dipole sets the characteristic size of particle-hole excitations, e.g. magnons in a ferromagnet, which in turn boosts their gap and stiffness. Indeed, the larger the particle-hole separation, the weaker the mutual attraction, and the stronger the excitation energy. In topological bands, this energy enhancement admits a lower bound within the local-mode approximation, highlighting the crucial role of topology in flat-band ferromagnetism. We illustrate these effects in microscopic models, emphasizing their generality and relevance to moiré materials. Our results establish the quantum-geometric dipole as a predictive geometric indicator for ferromagnetism in flat bands, a crucial prerequisite for topological order.

Figures (5)

• Figure 1: a) Schematics of the magnon dipole ${\bm d}$ measuring the distance between the $\downarrow$-particle ($p$, blue) and $\uparrow$-hole ($h$, orange) forming the flavor-flipping excitation. b) As the average dipole increases, the attraction between the oppositely charged $p$ and $h$ weakens, which decreases the magnon's binding energy (red) and increases its total energy (green) -- see Eq. \ref{['eq_magnoninteractiongap']}. The quantum-geometric dipole $\mathcal{S}^{\rm geom}$ sets the typical amplitude of the magnon dipole, resulting in larger magnon energies (non-shaded area) that can be lower bounded up to $\mathcal{O}(1)$ factors by topological invariants -- see Eqs. \ref{['eq_minimumstiffness']} and \ref{['eq_SMAgapbound']}.
• Figure 2: Magnon gap versus $M/A$ for the 2D BHZ model, shown in units of $U_0$. The magnon gap is plotted using three different methods: the Bethe–Salpeter analysis (SM Eq. \ref{['eq:bs']} blue); the local-mode approximation, with $z_{\bm k} = 1$ (SM Eq. \ref{['eq:sma_s']}, orange); and the geometric contribution formulas (Eq. \ref{['eq_magnoninteractiongap']}, magenta; Eq. \ref{['eq_SMAgap']}, red). The shaded blue (red) regions correspond to topological phases with Chern number $C=1$ ($C=-1$) for spin-$\uparrow$. The inset shows $d_{\rm min}^2 = \langle ||\mathcal{S}^{\mathrm{geom}}_{\bm k}||^2 \rangle_{|s|^2}$, the average squared quantum-geometric dipole. Parameters:$A = 1$, $B = 1$, and $r_{\xi} = 0.1$.
• Figure 3: Same as Fig. \ref{['fig:bhz']} for the continuum model of $\theta$-twisted bilayer MoTe$_2$. Gray dashed line marks the transition from polarized to unpolarized phases, $\Delta \varepsilon_{c}^{\rm exp} \approx 13~{\rm meV}$, extracted from the MCD measurements of Ref. cai2023signatures. Using a relaxed Stoner criterion, our theory estimates the transition point to occur at around $\Delta \varepsilon_{c}^{\rm th} \approx 16-17~{\rm meV}$, corresponding to the crossing points between the maximal kinetic energy gain $K_{\rm kin} = \max(|\varepsilon_{\bm k, \downarrow} - \varepsilon_{\bm k, \uparrow}|)$ (black line) and the magnon interaction energy (colored lines). Parameters:$\theta=3.7^{\circ}$, $\xi=30~\mathrm{nm}$, $\epsilon=7$, and $\xi_0=0.7~\mathrm{nm}$.
• Figure S1: Chern number of wavefunction $| u^{\uparrow}_{\bm k} \rangle$ as function of $M/B$
• Figure S2: (a) Berry curvature $\Omega(\mathbf{k})$ and (b) trace of quantum metric $\mathrm{Tr}g(\mathbf{k})$ for spin-$\uparrow$ with zero displacement field ($\Delta\varepsilon = 0$). (c) Maximum energy difference $K_{\rm kin} = \max(|\varepsilon_{\bm k, \downarrow} - \varepsilon_{\bm k, \uparrow}|)$, bandwidth of the top band, and band gap between the first and second highest bands versus the displacement potential $\Delta\varepsilon$. The gray dashed line marks the topological phase transition point.