Quantum geometry, localization, and topological bounds of spin fluctuations

Abstract

We study how topological crystalline defects--dislocations--reshape the real-space quantum geometric tensor and act as tunable sources of quantum geometry. We show that dislocations strongly enhance the quantum metric, establishing a direct link between lattice topology and the Hilbert-space geometry of states. We characterize the quantum geometry of topological magnons in ordered arrays of dislocations, demonstrating that defect-induced geometric enhancement controls their localization and topological protection. In disordered arrays, dislocation-driven geometry expands the accessible topological phase space and enables transitions to disorder-induced topological phases. Our results identify the quantum metric as a tunable bridge between crystalline topology, magnonic excitations, and emergent topological matter in aperiodic solid-state and synthetic systems.

Figures (3)

• Figure 1: (a) Magnonic band spectrum with $D=\Delta_{K}=0$ with the topological gap. (b) QM density in momentum space $\mathrm{Tr}[g_{\mu\nu}(\boldsymbol{k})]$ and Berry curvature $\Omega(\boldsymbol{k})$ for the case (a). Quantum metric $\mathcal{G}$, quantum volume $\mathcal{V}_{g}$ and Chern number $\mathcal{C}$ as a function of $\Delta_{K}$ (c) and DMI (d) for the lower band. In all plots, the parameters are set to $J=1$, $K=20J$ and $F=7J$.
• Figure 2: Bulk magnon structure of the magnetic triangular arrays of dislocations, along first BZ loop $\Gamma$-$M$-$K$-$\Gamma$, for the parameters $J=1$, $K=20J$, $F=7J$ and $\Delta_K=5J$. Inside the gap $\Delta_b$ lies a pair of magnonic states. The higher-energy mode of this pair is highlighted in blue, and zoomed-in for other parameter values.(b) Trace of the quantum metric $\mathcal{G}$, quantum volume $\mathcal{V}_{g}$ and Chern number $\mathcal{C}$, as a function of $\Delta_{K}$ for the lower band. (c) Real-space distribution of the magnonic mode highlighted in blue at (a), represented by $\Gamma_u(\boldsymbol{r})$, localized at the core of dislocations.
• Figure 3: Geometric and topological properties of magnonic states in the presence of randomly distributed dislocations on a hexagonal lattice. Examples of systems with low ($\rho=0.25$) and high ($\rho=0.6$) dislocation concentrations are shown in panels (a) and (b), respectively, for $L_x=L_y=XX$. Panel (c) characterizes the corresponding degree of crystalline disorder via the bond-orientational correlation function ${\cal G}_6(r)$ for different dislocation densities $\rho=0.1$, $0.25$, and $0.6$. At high concentrations, it exhibits a power-law decay, ${\cal G}_6(r)\sim r^{-\eta}$, with $\eta=2.5$ and $\eta=3.1$ for $\rho=0.25$ and $\rho=3.1$, respectively, signaling the emergence of a Hexatic phase. The quantum metric ${\cal G}$ and the Bott index ${\cal B}$ (for the lower band), evaluated as functions of the dislocation concentration and for $F/J=3$ and $5$, are shown in panels (e) and (f), respectively. The local quantum metric $g(r)$, Eq. (\ref{['eq:G_real_space']}), computed for $\rho=0.25$ and $\rho=0.6$, is displayed (in yellow) at panels (a) and (b), respectively. (d) Magnonic band spectrum determined for various dislocation concentrations. The transition to a topologically trivial phase (${\cal B}=0$) coincides with a pronounced enhancement of the quantum metric, indicating the formation of highly localized magnonic states.