Percolation from Quantum Metric in Flat-Band Delocalization

Abstract: The quantum metric is a fundamental ingredient of band quantum geometry and has recently at tracted intense interest, with most of its transport signatures appearing in the intrinsic second order nonlinear conductivity. In the clean limit, previous works argued that linear response conductivity is insensitive to the quantum metric, while the Berry curvature yields an intrinsic anomalous Hall con tribution. Here we combine analytic derivations with new numerics to show that disorder modifies the linear response conductivity dominated by geometric conductivity which is determined by the real space quantum metric. Focusing on a two dimensional multi-flatband stub-pyrochlore lattice, we identify a critical delocalized regime sandwiched between flat band localization and Anderson localization, characterized by finite geometric conductivity. Upon including spin orbit coupling, this regime evolves into a diffusive metallic phase, constituting a two dimensional inverse Anderson transition. Moreover, exploiting the connection between the real space quantum metric marker and the Wannier function spread, we construct a bond-percolation model on a square lattice. The resulting percolation region quantitatively coincides the critical delocalized regime, the exponent of which supports a classical percolation universality class. These findings suggest that flat band de localization can be understood as a classical percolation of quantum metric puddles. This advances our understanding of quantum geometric contributions to transport and establishes linear response measurements as a new avenue for accessing the quantum metric.

• The paper establishes that disorder in flat-band lattices induces delocalization via a percolation mechanism governed by the quantum metric.
• It employs self-consistent Born approximation, finite-size scaling, and recursive Green's function techniques to quantify geometric conductivity.
• The study demonstrates that Rashba spin-orbit coupling enhances quantum metric effects, offering new avenues for controlling disorder-induced metallicity.

Percolation from Quantum Metric in Flat-Band Delocalization

Introduction

The quantum metric, constituting the real part of the quantum geometric tensor, has emerged as a key quantity for understanding quantum geometry in band structures. While prior work established the Berry curvature's role in anomalous Hall contributions to linear transport, and the quantum metric's influence in intrinsic nonlinear responses, the direct connection between the quantum metric and linear conductivity in disordered flat-band systems had not been rigorously addressed. This work systematically elucidates how disorder-driven broadening in multi-flatband lattices enables a geometric mechanism for delocalization, quantitatively described by the real-space quantum metric and mapped onto a classical bond percolation process.

Model and Formalism

The study centers on a two-dimensional multi-flatband stub-pyrochlore lattice specifically constructed to exhibit two well-separated flat bands, deliberately excluding contributions from other geometric effects. The model incorporates nearest-neighbor hoppings, crossing couplings, and stub hybridizations, as shown in the schematic representation and associated band structure dispersion. Introduction of Rashba spin-orbit coupling provides a symmetry-controlled enhancement of quantum geometric effects.

Figure 1: (a) Schematic of a 2D stub-pyrochlore lattice, highlighting the relevant hopping parameters and localized orbitals; (b) Bands of the SOC-free Hamiltonian showing isolated flat bands; (c) Spinful extension with Rashba SOC retains flat bands.

The Hamiltonian is explicitly defined in momentum space, accommodating spinful extensions and disorder via a uniformly distributed onsite potential. The disorder strength $W$ induces spectral broadening $\eta$, self-consistently evaluated in the self-consistent Born approximation. Crucially, the geometric linear conductivity in the disorder window between flat bands is calculated by tracing the real-space quantum metric weighted by the energetically broadened interband transitions.

Numerical Results: Disorder and Geometric Conductance

Transport calculations utilize the Landauer-Buttiker formalism with recursive Green's function techniques. The study demonstrates that as disorder strength increases from the clean limit, three regimes emerge: a flat-band localized insulator, a critical quasi-metallic regime with enhanced geometric conductance, and an Anderson localized insulator at high disorder.

Figure 2: (a) $\eta$ versus $W$ for both SOC-free and Rashba-SOC models; (b) Conductance $G$ and geometric conductivity $\sigma_{\rm geo}$ as a function of $W$—with and without SOC—revealing conductance enhancement and critical behavior; (c) In the presence of SOC, a robust diffusive metallic phase appears; (d) Renormalized localization length finite-size scaling evidences two critical transitions.

In the flat-band localized regime, conductance is suppressed due to destructive interference. Upon increasing $W$, the conductance increases sharply, coinciding with the onset of spectral broadening. Notably, geometric conductivity $\sigma_{\rm geo}$—directly tied to the quantum metric—closely tracks the transport conductance, substantiating the geometric underpinning of the delocalization. When Rashba SOC is included, the critical regime broadens and conductance is enhanced, consistent with the strengthening of the quantum metric. Finite-size scaling of the localization length reveals two critical points, marking a two-dimensional inverse Anderson transition, with exponents near those of the symplectic universality class.

Geometric Percolation Framework

To interpret the transition, the work constructs a real-space percolation model based on the quantum metric marker. Each site is associated with the Wannier spread, obtained directly from the quantum metric, serving as a measure of the effective region in which an electron can delocalize due to quantum geometric effects.

Figure 3: Visualization of the geometric percolation concept, in which local quantum metric puddles (valleys) connect via tunneling to form system-spanning conductive paths as disorder lifts localization.

Neighboring valleys are connected if the combined local spreads exceed a threshold, forming "perfect links." The percolation threshold, controlled by disorder strength, dictates the transition from isolated puddles to a system-spanning conductive cluster.

Figure 4: Spanning probability $p_t(W)$ for various system sizes, demonstrating finite-size scaling consistent with classical percolation universality at both disorder-induced transitions.

The spanning probability $\eta$0, the likelihood that a conducting path connects opposite edges, exhibits a sharp transition as disorder is tuned, with critical exponents matching those of classical percolation (ν ≈ 1.33). Finite-size scaling analysis confirms the universality and robustness of the percolation picture, with the critical disorder thresholds coinciding with the transport-defined boundaries of localization and delocalization regimes.

Theoretical and Practical Implications

These results demonstrate that in multi-flatband systems with strong destructive interference, disorder can dynamically activate delocalization via a geometric mechanism fundamentally distinct from conventional Anderson transitions in dispersive bands. The direct equivalence of the geometric conductivity and transport conductance, combined with its critical behavior under finite-size scaling, supports the identification of a "quantum metric percolation" universality class. Notably, the conductivity can be tuned by symmetry breaking mechanisms such as spin-orbit coupling, offering a pathway for engineering disorder-induced metallicity in nominally insulating flat-band materials.

From a practical perspective, these findings establish linear-response transport as a measurable probe of quantum metric properties, complementing nonlinear or spectroscopic approaches. The percolation framework paves the way for predictive modeling of disorder-driven transitions in engineered flat-band systems, including quantum simulators and designer lattices.

Conclusion

This study provides a comprehensive analytic and numerical demonstration that disorder-driven delocalization in multi-flatband lattices is governed by a quantum metric-induced classical percolation of local quantum puddles. The geometric conductance, directly related to the real-space quantum metric, offers a quantifiable and tunable handle on critical transport phenomena. These results connect band geometry, disorder physics, and percolation theory, highlighting the central role of the quantum metric in flat-band systems and opening new opportunities for geometric control of quantum transport.