🔬 mesoscale physics

Aharanov-Bohm oscillations and perfectly transmitted mode in amorphous topological insulator nanowires

This study demonstrates that while amorphous topological insulator nanowires retain robust Aharanov-Bohm oscillations and a perfectly transmitted mode at low energies due to chiral or statistical time-reversal symmetry, high levels of amorphicity destroy these features, leading to nonquantized resonances associated with a transition to a trivial insulating phase.

Original authors: Miguel F. Martínez, Adolfo G. Grushin, Jens H. Bardarson

Published 2026-02-18

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CC BY 4.0

Original authors: Miguel F. Martínez, Adolfo G. Grushin, Jens H. Bardarson

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

Imagine you have a magical highway made of a special material called a Topological Insulator. This highway has a very strange rule: cars (electrons) can only drive on the very edge of the road, and they are locked into a specific lane. If they try to turn around or hit a bump, they simply cannot stop or bounce back; they are forced to keep moving forward. This is called a "perfectly transmitted mode."

Now, imagine you wrap this highway in a giant, invisible magnetic blanket (a magnetic flux). In a perfectly straight, crystal-clear highway, this blanket creates a rhythmic pattern: as you tighten or loosen the blanket, the traffic flow goes up and down in a perfect, predictable wave. At certain points, the traffic flows perfectly without a single car getting stuck.

The Problem: The "Amorphous" Mess
Real-world materials aren't perfect crystals. They are often "amorphous," which means they are like a pile of sand or a jumbled pile of LEGOs rather than a neat grid. The authors of this paper asked: What happens to our magical highway if the road itself is bumpy, wobbly, and irregular?

In a perfect crystal, the road is the same width everywhere. But in an amorphous wire, the road might be wide in one spot, narrow in another, and twisty throughout. The researchers were worried that this messiness would break the magic, causing cars to crash, bounce back, and ruin the perfect flow.

The Discovery: The Magic Survives (For a While)
The team ran computer simulations to see how this "jumbled" highway behaved. Here is what they found, using some simple analogies:

The "Statistical" Safety Net:
Even though the road is bumpy, the researchers found that for a while, the magic still works! It's like a crowded dance floor. Even if the floor is uneven, if the dancers (electrons) are moving in a specific way, they might not notice the bumps. The paper explains that the "jaggedness" of the road averages out. The road might be wide here and narrow there, but on average, it behaves like a smooth road. This "statistical" smoothness protects the perfect flow of traffic, allowing the cars to zip through without bouncing back, even in a messy wire.
The "Chiral" Superpower:
At very low energy (slow speeds), the cars have a special "superpower" called Chiral Symmetry. Think of this as a rule that says, "If you are moving forward, you must move forward; you are forbidden from turning around." Even if the road is a total mess, this rule is so strong that the cars simply ignore the bumps and keep going. This is why the perfect flow survives in the "messy" wires at low energies.
The Breaking Point (The "Trivial" Phase):
However, there is a limit. If the road gets too messy (high "amorphicity"), the magic breaks.
- The Analogy: Imagine the road becomes so twisted and narrow that it turns into a dead-end alley. The cars can't find a path through. Instead of a smooth flow, you get "traffic jams" or "echoes" (resonant peaks).
- The Result: The perfect, rhythmic wave of traffic disappears. Instead, the cars get stuck in specific spots, creating "bound states." The highway has effectively turned from a magical, topological road into a boring, regular insulator where nothing moves easily.

Why Does This Matter?
This research is crucial for the future of technology.

Better Electronics: We want to build tiny wires for computers that don't overheat. The "perfect flow" means no resistance and no heat. This paper tells us that we don't need perfect, expensive crystals to build these wires; we might be able to use cheaper, messier, amorphous materials and still get the same super-efficient performance.
Quantum Computers: These special wires are candidates for holding "quantum bits" (qubits). Knowing that the magic survives in messy materials makes it much easier to build real-world quantum computers, because we don't have to worry about making the materials perfectly smooth.

In a Nutshell:
The paper proves that the "magic" of topological insulators is surprisingly robust. Even if you build the wire out of a jumbled, messy material, the electrons can still find a way to flow perfectly without getting stuck—as long as the mess isn't too extreme. It's like finding out that a river can still flow smoothly even if the riverbed is made of rocks, as long as the rocks aren't piled up so high that they block the water completely.

1. Problem Statement

Topological insulator (TI) nanowires are promising candidates for next-generation interconnects and topological quantum computing due to their surface states, which are protected against backscattering by time-reversal symmetry (TRS) and topology. In crystalline nanowires, threading a magnetic flux ( $\phi$ ) through the cross-section induces Aharonov-Bohm (AB) oscillations in conductance. Crucially, at specific flux values ( $\phi = (2n+1)\phi_0/2$ ), an effective TRS is restored, leading to a perfectly transmitted mode (conductance $G = e^2/h$ ) that is immune to backscattering.

However, real-world materials often possess structural imperfections. While crystalline TIs have been extensively studied, the robustness of these topological transport signatures in amorphous (non-crystalline) nanowires remains unclear. Amorphous structures introduce two major challenges:

Broken Translation Invariance: The cross-section varies along the wire axis, potentially breaking the effective TRS required for the perfectly transmitted mode.
Disorder: Structural randomness could induce localization or destroy the topological phase entirely.

The central question addressed is: Under what conditions do AB oscillations and the perfectly transmitted mode survive in amorphous topological nanowires, and how does amorphicity drive a topological phase transition?

2. Methodology

The authors employ a numerical transport approach using a tight-binding model adapted for amorphous systems.

Model Hamiltonian: They adapt the Fu-Berg model for 3D TIs (specifically the $Bi_2Se_3$ family) to non-crystalline geometries. The Hamiltonian includes spin-1/2 degrees of freedom, $p_z$ orbitals, and position-dependent hopping ( $t$ ), spin-orbit coupling ( $\lambda$ ), and other parameters ( $\eta$ ).
Amorphicity Definitions: Two types of disorder are simulated:
1. Layered-Amorphous: Identical 2D amorphous layers stacked along the $z$ -axis. This preserves translation invariance along the wire, keeping the magnetic flux constant.
2. Fully-Amorphous: All three spatial coordinates ( $x, y, z$ ) are randomized around a reference cubic lattice. This breaks translation invariance in all directions, causing the cross-sectional area and local magnetic flux to vary along the wire.
Transport Calculation: The nanowire is connected to semi-infinite crystalline leads. The zero-bias conductance ( $G$ ) is calculated using the Landauer formula and the Kwant software package.
Topological Characterization: To link transport to topology, the authors use local topological markers (specifically the local chiral marker $\nu(r)$ ). This allows for the characterization of topological phases in systems lacking translation invariance, averaging over the bulk region to determine the topological invariant.
Symmetry Analysis: The study investigates the role of Chiral Symmetry (present at $E_F=0$ in the clean limit) and Statistical Time-Reversal Symmetry (where local flux fluctuations average out in long, wide wires) in protecting transport modes.

3. Key Contributions and Results

A. Layered-Amorphous Nanowires

Survival of Perfect Transmission: In the layered case, despite atomic disorder within layers, translation invariance along the $z$ -axis is maintained. Consequently, the effective TRS is restored at $\phi = (2n+1)\phi_0/2$ .
Robustness: A perfectly transmitted mode ( $G = e^2/h$ ) persists at low energies and up to moderate amorphicity ( $w \lesssim 0.2$ ).
Mechanism: The mode is protected by the effective TRS. As amorphicity increases, the conductance peaks broaden, and the flux value for perfect transmission shifts slightly due to finite-size effects and bulk penetration depth, but the quantization remains until the disorder becomes too strong.

B. Fully-Amorphous Nanowires

Symmetry Protection: In fully amorphous wires, the varying cross-section breaks exact TRS. However, the authors demonstrate that a perfectly transmitted mode can still exist via two mechanisms:
1. Chiral Symmetry: At $E_F = 0$ and in the absence of scalar disorder, the system belongs to symmetry class AIII, where chiral symmetry protects the mode.
2. Statistical TRS: For sufficiently long and wide wires, local flux fluctuations average out, effectively restoring TRS in a statistical sense.
Transition to Resonant Transport: As amorphicity increases beyond a critical threshold ( $w \sim 0.3 - 0.4$ $w \sim 0.3 - 0.4$ ):
- The perfectly transmitted mode disappears.
- Conductance oscillations vanish, replaced by sharp, non-quantized resonant peaks ( $G < e^2/h$ ).
- These peaks correspond to bound states (localized modes) within the surface gap.
- The conductance of these resonances decays exponentially with nanowire length ( $L$ ), confirming their localized nature.

C. Topological Phase Transition

Marker Analysis: Using the local chiral marker $\nu$ $ν$ , the authors identify a topological phase transition driven by amorphicity.
- Topological Phase ( $w < 0.3$ ): The bulk-averaged marker approaches the quantized value $\nu = -1$ . The local marker distribution shows a topological core ( $\nu \approx -1$ ) compensated by the boundary.
- Trivial/Indeterminate Phase ( $w > 0.4$ ): The marker deviates from quantization, approaching non-integer values (e.g., $\nu \approx -0.15$ or $0$). The distinction between bulk and boundary vanishes.
Correlation: The loss of the perfectly transmitted mode and the emergence of resonant bound states coincide with this transition from a topological phase to a trivial (or ill-defined) insulating phase. The localization length of the single-particle density matrix exceeds the cross-section size in the high-amorphicity regime.

4. Significance and Implications

Robustness of Topological Transport: The study confirms that topological transport signatures (AB oscillations and perfect transmission) are robust against structural disorder, provided the amorphicity is moderate and the system retains specific symmetries (Chiral or Statistical TRS).
Material Viability: This supports the feasibility of using amorphous topological materials (like amorphous $Bi_2Se_3$ ) for practical applications, such as low-resistance interconnects or topological qubits, as they are easier to fabricate than perfect crystals.
Diagnostic Tools: The paper establishes that the transition from quantized conductance to resonant bound states serves as a transport signature for a topological phase transition in disordered systems.
Theoretical Framework: It provides a rigorous framework for analyzing topological phases in amorphous systems using local markers and statistical symmetries, bridging the gap between idealized crystalline theory and disordered experimental realities.

In conclusion, the authors demonstrate that while strong amorphicity eventually destroys the topological protection of surface states in nanowires, a significant regime exists where amorphous TIs retain the unique transport properties of their crystalline counterparts, making them viable candidates for future quantum and nanoelectronic technologies.