🔬 mesoscale physics

Majorana polarization in disordered heterostructures

This paper demonstrates that while Majorana polarization is a useful indicator of topological features in disordered heterostructures, it is insufficient on its own to distinguish true Majorana bound states from trivial or partially separated Andreev bound states, necessitating a combined analysis of energy spectra, wavefunction localization, and topological bandgaps for reliable identification in topological quantum computation.

Original authors: Shubhanshu Karoliya, Sumanta Tewari, Gargee Sharma

Published 2026-08-07

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CC BY 4.0

Original authors: Shubhanshu Karoliya, Sumanta Tewari, Gargee Sharma

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 are a detective trying to find a very special, elusive creature hiding in a complex city. This creature, known in the scientific world as a "Majorana fermion," is a bit of a paradox: it is its own antiparticle. In the realm of solid-state physics, scientists have been building "cities" out of tiny wires and superconductors, hoping to trap these creatures at the very edges. Why? Because these creatures are the keys to building a new kind of computer—one that is incredibly powerful and doesn't crash easily, known as a topological quantum computer.

To catch these creatures, scientists have been using a specific "trap" or diagnostic tool called "Majorana polarization." Think of this like a special flashlight that is supposed to glow only when the real creature is present. The idea was that if you shine this flashlight on the left side of the wire and the right side, and the lights dance in a specific, opposite pattern, you have found your prize. However, just like in a real city, there are lookalikes. There are "quasi-Majorana" modes—imposters that look almost exactly like the real thing but are actually just ordinary, messy states caused by the dirt and disorder in the wire. The big question has been: Can our special flashlight tell the difference between the real creature and the imposter, especially when the city is messy?

This paper, written by Shubhanshu Karoliya, Sumanta Tewari, and Gargee Sharma, dives deep into this mystery. They simulated two different types of "cities": a simple, one-dimensional nanowire and a slightly more complex, quasi-one-dimensional system. Both were set up with the necessary ingredients to potentially host Majorana creatures: spin-orbit coupling (a fancy way of saying the electrons spin as they move), superconductivity, and a magnetic field. Crucially, they added "disorder"—random bumps and impurities—to make the simulation realistic, because real wires are never perfectly clean.

The authors found that the "flashlight" of Majorana polarization is not as reliable as we hoped. In their simulations, they discovered scenarios where the polarization lights up perfectly, showing the expected opposite pattern between the left and right sides of the wire, even when the real Majorana creature is not there. Instead, the wire is hosting a "quasi-Majorana" or a "partially separated Andreev bound state." These are the imposters. They look like the real thing to the polarization test, but they lack the two other critical features needed for a true topological quantum computer: a clean energy gap in the middle of the wire and wavefunctions that are strictly locked to the very ends.

The researchers showed that in disordered systems, you can have a situation where the polarization product is exactly what you want (close to -1), yet the system is actually in a "trivial" phase, meaning it's just a messy, ordinary state. They also found that even in the topological phase, where the real creatures should be, the polarization can sometimes behave strangely or fail to show the expected pattern if the disorder is just right. This happens regardless of which mathematical definition of polarization you use.

The takeaway from this study is a cautionary one for the field. Relying solely on Majorana polarization to confirm the existence of these topological states is risky. The authors suggest that to be truly sure you have found a Majorana bound state useful for quantum computing, you need to use a combination of tools. You must check the polarization, yes, but you also need to look at the energy spectrum to ensure there is a proper gap, and you must verify that the wavefunctions are actually localized at the edges. In the messy, disordered world of real materials, one clue is never enough; you need the whole picture to catch the real creature and avoid the imposters.

Technical Summary: Majorana Polarization in Disordered Heterostructures

Problem Statement
The identification of topological Majorana bound states (MBS) in engineered semiconductor-superconductor heterostructures remains a central challenge in condensed matter physics, particularly due to the prevalence of topologically trivial states that mimic MBS signatures. Specifically, in disordered systems, partially separated Andreev bound states (psABS), also known as quasi-Majorana modes, can produce zero-bias conductance peaks indistinguishable from true MBS in local measurements. While Majorana polarization ( $P$ ) has been proposed as a diagnostic tool to distinguish true topological MBS from trivial zero-energy states, its reliability in the presence of disorder is unclear. Previous literature suggests that true topological MBS satisfy the non-local correlation condition $P^*_{\text{left}} \cdot P_{\text{right}} \sim -1$ . This paper investigates whether this condition is sufficient to uniquely identify topological MBS or if it can be satisfied by trivial psABS in disordered systems.

Methodology
The authors analyze two distinct systems subject to Rashba spin-orbit coupling, proximity-induced superconductivity, external Zeeman fields, and disorder:

One-Dimensional (1D) Nanowire: A chiral-symmetric system modeled by a tight-binding Hamiltonian with nearest-neighbor hopping, Zeeman splitting, Rashba coupling, and induced pairing. Disorder is modeled via randomly distributed short-range impurities.
Quasi-One-Dimensional (Quasi-1D) System: A generic two-dimensional semiconductor-superconductor hybrid structure consisting of five coupled chains. This system breaks chiral symmetry due to transverse Rashba spin-orbit coupling terms.

The study employs numerical exact diagonalization of the Bogoliubov-de Gennes (BdG) Hamiltonian. The authors evaluate Majorana polarization using two distinct frameworks:

Chiral Definition: Applied to the 1D wire, defining polarization based on the difference in probabilities of chiral Majorana components ( $\gamma_A$ and $\gamma_B$ ).
Particle-Hole Definition: Applied to the quasi-1D system, defining polarization as the expectation value of the particle-hole operator $\mathcal{C}$ , normalized over spatial regions.

The analysis focuses on the behavior of the polarization product $P^*_{\text{left}} \cdot P_{\text{right}}$ , the zero-energy splitting ( $\delta E$ ), the topological bandgap ( $\xi$ ), and the spatial localization of wavefunctions across varying Zeeman fields and chemical potentials.

Key Contributions and Results
The study reveals that Majorana polarization, when used as a standalone diagnostic, is insufficient to reliably distinguish true topological MBS from trivial psABS in disordered environments.

Failure of the $P^*_{\text{left}} \cdot P_{\text{right}} \sim -1$ Criterion: The authors identify specific scenarios in both 1D and quasi-1D disordered systems where the condition $P^*_{\text{left}} \cdot P_{\text{right}} \sim -1$ is satisfied. However, in these instances, the system is in a topologically trivial phase characterized by the absence of a bulk topological bandgap and the lack of wavefunction localization at both wire edges. Instead, these states correspond to psABS where one Majorana component is localized at an edge while the other is only partially separated (localized near the middle or the same edge).
Sensitivity to Disorder and Overlap: In disordered 1D wires, the polarization magnitude is found to be highly sensitive to the overlap between Majorana modes rather than strictly proportional to the energy splitting, a relationship that is monotonic in clean systems but non-monotonic in disordered ones. Consequently, psABS can exhibit polarization magnitudes approaching unity, mimicking true MBS.
Quasi-1D System Behavior: In the quasi-1D system, the particle-hole definition of polarization shows that in the trivial phase, the polarization magnitudes $|P_{\text{left}}|$ and $|P_{\text{right}}|$ can reach values close to unity, and the product $P^*_{\text{left}} \cdot P_{\text{right}}$ can approach $-1$ even before the topological phase transition occurs. This contrasts with clean systems where polarization vanishes in the trivial phase.
Robustness Across Definitions: These conclusions hold regardless of whether the chiral framework (1D) or the particle-hole framework (quasi-1D) is used to define Majorana polarization.

Significance and Claims
The paper asserts that the condition $P^*_{\text{left}} \cdot P_{\text{right}} \sim -1$ is necessary but not sufficient for the identification of topological MBS suitable for topological quantum computation (TQC). The authors emphasize that robust identification requires a multifaceted approach combining:

Majorana polarization diagnostics.
Analysis of the energy spectra (specifically the presence of a topological bandgap).
Verification of wavefunction localization at the edges.

The study concludes that relying solely on polarization metrics can lead to false positives in disordered systems, where trivial psABS mimic the signatures of topological states. Therefore, a combination of complementary diagnostics is essential for confirming the topological nature of zero-energy modes in solid-state heterostructures.

Technical Summary: Majorana Polarization in Disordered Heterostructures

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