The Bott Metric: A Real-Space Bridge Between Topology and Quantum Metric

Abstract

The Bott index has become an indispensable tool to probe the topology of quantum matter, particularly in systems lacking translational symmetry. Constructed from a plaquette operator, it retains the phase information while discarding the amplitude. Here we introduce and develop the Bott metric, which captures this complementary amplitude information and provides a measure of the underlying quantum metric of the system. We show that, in the thermodynamic limit, the Bott metric converges to the trace of the integrated quantum metric. Our framework provides a new route to reveal the quantum metric structure in non-periodic systems, which we illustrate using representative examples ranging from disordered to amorphous models. More broadly, our definition of the Bott metric unifies the notion of topological invariants and quantum metric under the same overarching plaquette operator construction.

Figures (3)

• Figure 1: Plaquette operator framework leading to the Bott metric. (a) Plaquette formed by the torus twists $U$, $V$, $U^\dagger$, and $V^\dagger$ in twist-angle space. While the unprojected loop closes trivially, its compression to the occupied subspace produces a nontrivial response. (b) A single projected step on the occupied subspace $P\mathcal{H}$. Acting with a twist operator (shown here as $V$) generally pushes a normalized occupied state $|\hat{\psi}\rangle$ slightly outside $P\mathcal{H}$. Projecting back removes the component in $\mathcal{Q}\mathcal{H}$, whose norm $d_{\mathrm{ch}}=\|\mathcal{Q} V|\hat{\psi}\rangle\|$ is the one-step leakage. In the small-twist limit, this leakage gives the elementary contribution to the quantum distance. (c) Summary of the full construction. From the torus twists $U=e^{i\theta_x}$ and $V=e^{i\theta_y}$, with $\theta_x=2\pi X/L$ and $\theta_y=2\pi Y/L$, we form the projected-and-extended operators $\widetilde{U}=\mathcal{Q}+PUP$ and $\widetilde{V}=\mathcal{Q}+PVP$, and the plaquette operator $W=\widetilde{U}\widetilde{V}\widetilde{U}^\dagger\widetilde{V}^\dagger$. The quantity $\mathrm{Tr}\log(W)$ then yields two complementary diagnostics: its imaginary part gives the Bott index, while its real part, introduced below as the Bott metric, measures the total loop contraction and, in the localized regime, yields the integrated quantum metric.
• Figure 2: Bott metric in clean and disordered Qi-Wu-Zhang model.(a) The Chern number $C$, the Bott metric $\mathcal{M}_b$, and the integrated quantum metric $\mathrm{Tr}(G)$ as a function of the mass $m$ for the clean Qi-Wu-Zhang model. Here $L=30$ torus was used at half filling ($E_F=0$), with $A=1$ and $B=0.5$. (b) The disorder-averaged Bott index $\langle B\rangle$ over the $(m,W)$ plane on an $L=16$ torus with $t=\nu=1$ and $\gamma=0$ for the disordered Qi-Wu-Zhang model with disorder strength $W$. (c) Disorder-averaged Bott metric $\langle \mathcal{M}_b\rangle$ and (d) integrated quantum metric $\langle \mathrm{Tr}(G)\rangle$ for the same $(m,W)$ scan. In both the clean and the disordered phase diagrams, $\mathcal{M}_b$ closely tracks $\mathrm{Tr}(G)$: the curves in (a) are nearly indistinguishable and, in (c) and (d), both the moderate valued magenta colored region inside the quantized-$\langle B\rangle$ plateau and the bright yellow boundary band occur at the same $(m,W)$ locations and have identical color gradients, demonstrating the equivalence between the Bott metric and the trace of integrated quantum metric.
• Figure 3: Bott metric in amorphous systems.(a) Representative realization of an amorphous model with a hopping cutoff radius $R$. (b) Ensemble-averaged Bott index $\langle B\rangle$ and the Bott metric $\langle \mathcal{M}_b\rangle$ versus the mass parameter $M$. The shaded portions indicate the standard error of the mean obtained from the ensemble averaging. While $\langle B\rangle$ identifies a broad topological plateau, $\langle \mathcal{M}_b\rangle$ varies strongly across the same window and exhibits an asymmetric peak structure at the two topological phase transition points, revealing additional localization-sensitive geometric information beyond the quantized topological label. Here we set $L_x=L_y=25$, $\rho=0.6$, $R=4.0$, $t_2=0.25$, and $\lambda=0.5$.