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.

• The paper introduces the Bott metric, a new real-space measure derived from the plaquette operator that rigorously connects the Bott index to the integrated quantum metric.
• It benchmarks the metric against standard techniques in clean and disordered models, showing strong quantitative agreement and sensitivity to localization transitions.
• The method expands real-space diagnostics to amorphous, aperiodic, and non-Hermitian systems while offering computational efficiency and deeper insights into quantum phase transitions.

The Bott Metric: Real-Space Quantum Geometry from Plaquette Operators

Introduction and Motivation

The interplay between quantum topology and quantum geometry, particularly within systems lacking translational symmetry, constitutes a major frontier in condensed matter physics. The Bott index, a real-space topological invariant rooted in the phase winding of the so-called plaquette operator, has become a central diagnostic for topology in both crystalline and non-crystalline systems. However, the quantum metric—a quantity encoding the real part of the quantum geometric tensor—remained elusive in previous real-space approaches. This paper introduces the Bott metric, a real-space quantity constructed from the plaquette operator's amplitude, establishing a direct, rigorous connection to the integrated quantum metric (IQM) in the thermodynamic limit. The framework thus unifies topological and geometric diagnostics within the same operator-theoretic construction, expanding the utility of real-space approaches to quantum matter (2604.04447).

Formalism and Theoretical Foundation

The Bott metric is defined analogously to the Bott index but exploits the amplitude (norm contraction) rather than the phase (winding) of the plaquette operator:

• Let $P$ denote the projector onto occupied states (below the Fermi energy) and $U, V$ real-space phase twist operators, which encode position-dependent boundary condition twisting in $x$ and $y$ on a toroidal finite system.
• After extending the projected twists ($U_P = PUP$, etc.) to the full Hilbert space, the key object is the plaquette operator $W_P = U_P V_P U_P^\dagger V_P^\dagger$.
• While the Bott index is given by $\frac{1}{2\pi} \mathrm{Im} \; \mathrm{Tr} \log W$ (capturing quantized topology), the Bott metric is defined by

$$M = -\frac{1}{2\pi} \mathrm{Re} \; \mathrm{Tr} \log W,$$

measuring the total loop-induced norm contraction within the occupied subspace.

A crucial result is the explicit correspondence between $M$ and the trace of the integrated quantum metric tensor $G_{\alpha\beta}$, which quantifies the real-space averaged quantum distance induced by position operators. In the thermodynamic limit and in the presence of locality (i.e., gapped or Anderson localized/projector-local regimes):

$U, V$0

This provides an efficient and robust means to probe the quantum metric in real space, without recourse to translational symmetry or momentum-space band structure.

Numerical Results and Case Studies

The Bott metric is systematically benchmarked against the standard IQM trace in clean and disordered models, substantiating its theoretical validity.

Clean and Disordered QWZ Models

In the translationally-invariant Qi-Wu-Zhang model, $U, V$1 and $U, V$2 exhibit strong quantitative agreement. Both display sharp peaks at topological phase transitions (i.e., mass parameters $U, V$3) reflecting gap closings, and track the Chern number jumps precisely. In the presence of disorder, Bott metric and IQM remain in lockstep throughout the mobility-gap regions, with both quantities revealing pronounced structure at the phase boundaries—specifically, ridges associated with reduced localization and phase transitions—thus confirming the real-space quantum metric interpretation persists under strong disorder.

Amorphous Chern Insulators

Application to the amorphous Chern insulator model further demonstrates the practical importance of $U, V$4 as a localization-sensitive geometric probe in systems devoid of momentum space. While the Bott index identifies a broad topological plateau, $U, V$5 reveals significant variation and an asymmetric peak-dip structure across the same plateau. This variation is directly linked to changes in the localization properties and the breakdown of the gap, highlighting differences between phase boundaries that the quantized Bott index alone cannot resolve.

Conceptual and Practical Implications

The Bott metric establishes quantum metric and topology as complementary spectral characteristics of real-space plaquette operators. The approach defines a computationally negligible overhead relative to standard Bott index calculations since both metrics are derived from the same matrix logarithm. The construction is broadly applicable: it extends to disordered, aperiodic, amorphous, non-Hermitian, and higher-dimensional systems, wherever the Bott index is relevant (2604.04447).

Theoretically, this approach provides a rigorous bridge between noncommutative geometry and quantum geometric quantities, potentially informing future connections to Wilson loops, gauge-invariant descriptors, and localization diagnostics. The metric also serves as a tool for studying quantum phase transitions in the absence of Bloch states, and can be immediately deployed for physical interpretation of response properties (e.g., quantum metric contributions to superfluid weight) in numerically tractable, real-space settings.

Conclusion

The Bott metric is a robust, real-space measure of the integrated quantum metric, derived directly from the modulus of the plaquette operator within the Bott index framework. It extends geometric diagnostics to the non-crystalline regime, faithfully tracks the quantum metric under both clean and disordered conditions, and reveals new structure at topological and localization transitions absent in topological indices. This unified framework significantly strengthens the bridge between topology and quantum metric, and opens new directions for studying quantum geometry in complex, non-periodic materials (2604.04447).