🔬 mesoscale physics

Projected amorphous topological insulators

This paper introduces projected amorphous topological insulators (PATIs), a new class of systems where a fraction of randomly selected disconnected sites from a parent crystal retains quantized global or fragile local topological invariants and boundary modes, exhibiting a tunable strong-to-fragile quantum phase transition characterized by a specific correlation length exponent.

Original authors: Archisman Panigrahi, Bitan Roy

Published 2026-09-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Archisman Panigrahi, Bitan Roy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Materials that conduct electricity are familiar to us, but there is a more subtle class of matter known as insulators, which normally block the flow of current. For decades, physicists have been fascinated by a special subset of these insulators called topological insulators. In these materials, the interior remains a perfect insulator, yet the surface or edges conduct electricity with remarkable ease. This behavior is not accidental; it is protected by the underlying geometry of the material's atomic structure, making the conducting edges incredibly robust against impurities or damage. Usually, this protection relies on the atoms being arranged in a perfect, repeating crystal pattern, much like soldiers standing in a precise grid. The question that has long intrigued scientists is whether such robust, topological behavior can survive if that perfect order is destroyed, leaving the atoms scattered randomly as they are in glass or other amorphous solids.

A new study introduces a method to create these disordered topological materials, revealing that they can exist in two distinct forms: one that is globally robust and another that is surprisingly fragile. The researchers started with a standard, ordered crystal model known to be a topological insulator. Instead of trying to build a new material from scratch, they took a mathematical approach to "project" a new system from the old one. They imagined selecting a small, random fraction of the atoms from the original crystal grid, ensuring that none of these selected atoms were directly connected to each other by the usual short-range links. At this initial stage, these isolated atoms were just ordinary, non-conducting points with no special properties.

The crucial step came next. The researchers mathematically removed the rest of the crystal—the atoms they did not select—and calculated how their absence would change the physics of the remaining isolated points. This process, known as integrating out the missing sites, effectively rewired the system. It forced the remaining isolated atoms to communicate with one another through long-range connections that did not exist before. The result was a new, effective material made entirely of the randomly scattered points, but now connected by these new, long-distance links. The researchers call this a "projected amorphous brane."

When they analyzed this new, disordered material, they found it could behave as a topological insulator, but only under specific conditions. If the concentration of selected atoms was high enough, the system formed what they term a "strong" projected amorphous topological insulator. In this state, the entire material possessed a quantized global property that guaranteed the existence of conducting channels along its edges, even though the interior remained insulating. This confirmed that topological order could indeed emerge from a completely disordered arrangement of atoms, provided the right mathematical projection was applied.

However, the study uncovered a more subtle and unexpected phenomenon when the concentration of atoms was lowered. Below a certain critical threshold, the material entered a "fragile" phase. In this state, the global measure of the material's topology vanished, suggesting the system was no longer a topological insulator in the traditional sense. Yet, a closer look revealed that a small fraction of the individual atoms still displayed a local, quantized topological signature. It was as if the topological order had not disappeared entirely but had become fragmented, surviving only on a few scattered islands within the disordered sea. Despite this fragility, these systems still supported conducting modes along their boundaries, blurring the line between a true topological insulator and a normal one.

The researchers also identified a third state, a "projected amorphous normal insulator," which appeared when the underlying parameters of the original crystal were not topological to begin with. In this case, the projection process failed to generate any topological features, leaving a material that was insulating everywhere, with no conducting edges. By mapping out these different phases, the team showed that the transition between the strong and fragile states was not random but followed a precise mathematical scaling law. They determined that the distance between the atoms and the size of the energy gap in the material were linked in a universal way, suggesting that this behavior is a fundamental property of how topology can survive in disordered systems.

While these findings are currently based on computer simulations and mathematical models rather than physical experiments, they offer a new blueprint for understanding how order can emerge from chaos. The authors suggest that while creating such materials directly in quantum matter might be challenging due to the need for specific long-range connections, the principles could be tested in highly controllable laboratory setups. These include specialized electrical circuits, photonic lattices made of light, or sound-based structures, where the rules of the game can be tuned with high precision. The work demonstrates that the robust edge states characteristic of topological materials are not solely the domain of perfect crystals, but can also be engineered into the disordered, amorphous structures that make up much of the natural world.

Technical Summary: Projected Amorphous Topological Insulators

Problem and Motivation
The classification of insulators traditionally relies on crystalline order, where topological phases arise from band inversions at high-symmetry points in the Brillouin zone, protected by discrete translational and rotational symmetries. While topological insulators (TIs) have been extensively studied in crystalline systems, their realization in amorphous lattices—where sites are randomly distributed and lack long-range order—has been limited. Previous studies on amorphous lattices have predominantly identified only "translationally inert" topological phases (those originating from the $\Gamma$ point). A significant gap remains in understanding whether "translationally active" phases (originating from finite momentum points like $M$ ) and other complex topological phenomena can emerge in the absence of crystalline symmetry. This work addresses the challenge of realizing a broader catalog of topological phases, including both active and inert types, on amorphous structures.

Methodology: Projected Amorphous Branes (PABs)
The authors introduce a construction method termed "Projected Amorphous Topological Insulators" (PATIs). The core of this methodology involves:

Site Selection: A fraction ( $x$ ) of sites is randomly selected from a parent crystalline lattice (specifically a square lattice). Crucially, these sites are chosen such that they are not directly connected by the short-range hopping amplitudes of the parent tight-binding Hamiltonian. At this "bare" level, the system consists of localized atomic orbitals with no topology.
Projection/Integration: The remaining sites of the parent lattice are integrated out using a path integral approach. This procedure generates an effective Hamiltonian ( $H_{\text{eff}}$ ) for the selected amorphous subset.
Effective Hamiltonian: The resulting $H_{\text{eff}}$ is given by $H_{\text{eff}} = H_{11} - H_{12}H_{22}^{-1}H_{21}$ , where $H_{11}$ represents the bare Hamiltonian of the selected sites, and the second term introduces long-range hopping mediated by the integrated-out sites. This renormalized Hamiltonian couples all sites of the amorphous lattice, effectively endowing the disordered system with the topological characteristics of the parent crystal.
Model System: The authors apply this framework to the Qi-Wu-Zhang model (a time-reversal symmetry breaking insulator on a square lattice) to demonstrate the emergence of translationally active and inert phases.

Key Contributions and Results
The study identifies three distinct phases within the projected amorphous brane (PAB) system, characterized by the concentration of sites ( $x$ ) and the parent model's parameters:

Strong PATIs: Within the topological regime of the parent model, there exists a critical concentration $x_c$ above which the system forms a "Strong PATI."
- Characteristics: These phases exhibit a quantized global topological invariant, specifically the Bott index ( $\langle B \rangle = \pm 1$ ), and a quantized local topological marker (Local Chern Marker, LCM) on a large fraction of sites.
- Types: Depending on the parent model parameters, these can be "translationally active" (originating from the $M$ point) or "translationally inert" (originating from the $\Gamma$ point).
- Boundary Modes: They support robust, gapless boundary modes in systems with open boundary conditions (OBCs), consistent with the bulk-boundary correspondence.
Fragile PATIs: Below the critical concentration ( $x < x_c$ ) but still within the parent model's topological regime, a novel "Fragile PATI" phase emerges.
- Characteristics: The global topological invariant (Bott index) vanishes ( $\langle B \rangle = 0$ ). However, a small fraction of sites displays an almost quantized local topological marker.
- Boundary Modes: Despite the vanishing global invariant, these systems still host in-gap modes localized near the boundary, though the number of topological sites is significantly reduced compared to the strong phase.
Projected Amorphous Normal Insulators (PANIs): In the trivial parameter regime of the parent model (for any $x$ ), the system behaves as a normal insulator. Both the global invariant and local markers vanish, and no gapless boundary modes are present.

Quantum Phase Transition (QPT)
The transition between Strong and Fragile PATIs is a structural quantum phase transition tuned by the site concentration $x$ .

Scaling: At the critical point $x_c$ , the average inter-site distance $\xi$ exhibits universal scaling with the band gap $\Delta$ of the parent model.
Critical Exponents: The transition is characterized by a correlation length exponent $\nu$ . The authors find $\nu \in (1.00, 1.46)$ , which appears to depend on the dimensionality but not the symmetry class. This value is distinct from transitions driven by on-site disorder (where $\nu \approx 2.3 - 2.6$ ).
Data Collapse: The configuration-averaged Bott index displays single-parameter scaling behavior near $x_c$ , confirming the nature of the transition.

Significance and Claims
The paper claims to demonstrate a generic mechanism for realizing diverse topological phases on amorphous lattices without requiring crystalline order.

Breaking Limitations: It circumvents the limitation of previous amorphous TI studies by showing that translationally active phases can exist in amorphous systems via projection.
Novel Phase: It introduces the concept of "Fragile PATIs," a phase where global topology vanishes but local topological signatures persist, challenging the strict reliance on global invariants for identifying topological matter in disordered systems.
Universality: The results suggest that projected topological branes can capture a wide variety of lattice-based topological phases on geometric descendants, including quasicrystals, fractals, and amorphous lattices.
Experimental Outlook: While the required long-range hopping makes immediate realization in quantum materials challenging, the authors suggest that highly tunable classical metamaterials (topoelectric circuits, photonic, and phononic lattices) offer a promising avenue for experimental realization. The work also posits that fragile amorphous topological phases may be recognizable across all Altland-Zirnbauer symmetry classes.

Technical Summary: Projected Amorphous Topological Insulators

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