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Nonlocal Majorana polarization in non-Hermitian topological superconductors

This paper extends the concept of nonlocal Majorana polarization to non-Hermitian topological superconductors by incorporating biorthogonal eigenstates, demonstrating its ability to distinguish Majorana zero modes from trivial states and exceptional points while revealing that non-Hermiticity enhances the robustness of these modes.

Original authors: Arjun S. Kumar, Jorge Cayao, Oladunjoye A. Awoga

Published 2026-07-29

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CC BY 4.0

Original authors: Arjun S. Kumar, Jorge Cayao, Oladunjoye A. Awoga

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 a world where the rules of physics get a little fuzzy, not because things are moving too fast or are too small, but because the system itself is leaking energy or gaining it from the outside. This is the realm of non-Hermitian physics, a corner of science that studies materials and devices that aren't perfectly closed off. In this world, scientists are hunting for a very special kind of particle called a "Majorana zero mode." Think of these not as tiny balls, but as ghostly, self-conjugating twins that live at the very edges of a superconducting wire. They are famous for being incredibly stable and for holding information in a way that could one day power super-secure quantum computers. However, there's a catch: sometimes, the wire tricks you. It can host "trivial" zero-energy states that look exactly like the real ghosts but are actually just ordinary, fragile imposters. Distinguishing the real deal from the fake is the holy grail of this field.

For a long time, scientists had a reliable flashlight to find these real ghosts in perfect, closed systems. They called it "Majorana polarization," a way to measure how much the ghost is split between the two ends of the wire. If the measurement was perfect, the ghost was real and safe. But what happens when the system isn't perfect? What if the wire is leaking energy or interacting with its environment in weird ways? That's the big question this new paper tackles. The authors wondered if their old flashlight still works when the physics gets messy and non-Hermitian, or if they need a new tool to see through the fog.

In this study, the researchers took that old flashlight and upgraded it for the messy, non-Hermitian world. They developed a new method called "nonlocal Majorana polarization" that uses a special mathematical trick called "biorthogonal states" to handle systems where energy isn't conserved in the usual way. By testing this new tool on two different types of theoretical superconducting chains, they found something surprising and exciting. Their simulations show that this upgraded polarization doesn't just find the real Majorana ghosts; it can also tell them apart from the imposters (trivial states) and from a weird phenomenon called "exceptional points," where the system's rules break down completely.

Perhaps the most playful discovery is that the very thing that makes these systems "messy"—the non-Hermitian effects—actually makes the real Majorana ghosts stronger and more robust. In the perfect, closed world, adding a little chemical imbalance often made the ghosts wobble and disappear. But in this new, leaky world, the authors found that the non-Hermitian effects act like a stabilizing force, pushing the ghosts back to their perfect, zero-energy state and making them harder to knock out of place. They also introduced a "sensitivity" meter to measure exactly how much of this stability comes from the non-Hermitian effects.

The paper confirms that this new tool is a reliable way to map out the topological phases of these strange superconductors. Whether the system is a simple chain or a junction where a superconductor meets a normal metal, the nonlocal biorthogonal polarization successfully identifies the true Majorana modes. The authors demonstrate that while the system might look chaotic, the underlying topology is actually more resilient than we thought, provided we use the right lens to look at it. This suggests that in future experiments, where real-world wires are never perfectly isolated, we might actually find that the "imperfections" help protect the quantum information we are trying to store.

Technical Summary: Nonlocal Majorana Polarization in Non-Hermitian Topological Superconductors

Problem Statement
Majorana zero modes (MZMs) are self-conjugated quasiparticles in topological superconductors with potential applications in quantum computing. While their identification in Hermitian systems is well-established using the nonlocal Majorana polarization (MP)—a real-space topological indicator that distinguishes true MZMs from trivial zero-energy states (TZESs)—this framework has not been extended to non-Hermitian (NH) systems. NH effects, arising from gain/loss, coupling to reservoirs, or nonreciprocity, introduce complex spectra and exceptional points (EPs) where eigenvalues and eigenvectors coalesce. These features break the standard bulk-boundary correspondence and complicate the characterization of spatial nonlocality. The central problem addressed is whether the nonlocal MP can serve as a reliable diagnostic tool in NH superconductors to distinguish between MZMs, EPs, and TZESs, and how non-Hermiticity affects the robustness of these states.

Methodology
The authors generalize the concept of nonlocal MP to the non-Hermitian realm by incorporating the biorthogonal nature of NH eigenstates. In NH systems, the Hamiltonian $H \neq H^\dagger$ , necessitating distinct left ( $L$ ) and right ( $R$ ) eigenvectors.

Biorthogonal Definition: The local MP at site $j$ is defined as the expectation value of the particle-hole operator $C$ using a combination of left and right eigenvectors of the lowest energy state: $p_{j}^{ab} = \langle \Psi_j^{a} | C | \Psi_j^{b} \rangle / \langle \Psi_j^{a} | \Psi_j^{b} \rangle$ , where $a,b \in \{L, R\}$ .
Nonlocal Indicators: The authors define nonlocal MPs for the left ( $P_l^{ab}$ $P_{l}^{ab}$ ) and right ( $P_r^{ab}$ $P_{r}^{ab}$ ) halves of the system. The total nonlocal MP is $P^{ab} = P_l^{ab} (P_r^{ab})^*$ $P^{ab} = P_{l}^{ab} (P_{r}^{ab})^{*}$ .
- $P^{aa}$ (orthogonal) corresponds to the Hermitian limit.
- $P^{ab}$ with $a \neq b$ (biorthogonal) captures the unique NH contributions.
Sensitivity Metric: A new quantity, the nonlocal MP sensitivity $R$ $R$ , is introduced as the determinant of the MP matrix in the biorthogonal basis: $R = P_{LL}P_{RR} - P_{LR}P_{RL}$ $R = P_{LL} P_{R R} - P_{L R} P_{R L}$ . This metric quantifies the contribution of non-Hermiticity to the polarization.
- $R = -1$ indicates pure NH contribution (biorthogonal dominance).
- $R = 0$ indicates pure Hermitian contribution.
- $R = \pm \infty$ or NaN signals an Exceptional Point.
Models: The framework is applied to two representative models:
- The Hatano-Nelson-Kitaev (HNK) chain (spinless fermions with nonreciprocal hopping).
- The Non-Hermitian Rashba superconducting chain (spinful fermions with nonreciprocal hopping and spin-orbit coupling).
- Both models are analyzed as isolated chains and as superconductor-normal (SN) metal junctions.

Key Results

Distinction of States: The nonlocal biorthogonal MP ( $P^{LR}$ $P^{L R}$ or $P^{RL}$ $P^{R L}$ ) successfully distinguishes between different zero-energy states:
- MZMs: Characterized by $P^{LR} \approx -1$ and $R \approx -1$ , indicating robust, fully nonlocal, zero-energy modes.
- Trivial Zero-Energy States (TZESs): Characterized by $P^{LR} \approx 0$ and $R \approx 0$ , even if they possess zero real energy.
- Exceptional Points (EPs): Characterized by $P^{ab}$ and $R$ diverging to $\pm \infty$ or becoming NaN, as the denominator in the expectation value vanishes.
Enhanced Robustness: A primary finding is that non-Hermiticity (specifically nonreciprocity $\Gamma$ $Γ$ ) enhances the robustness of MZMs.
- In the "sweet spot" (exact zero energy) of minimal chains (2-site and 3-site HNK), increasing $\Gamma$ drives the system toward $P^{LR} = -1$ and $R = -1$ , even when a finite chemical potential ( $\mu$ ) would otherwise degrade MZM quality in Hermitian systems.
- In longer chains and Rashba models, nonreciprocity reduces the energy splitting of Majorana states and pushes the system toward the topological phase, effectively restoring high-quality MZMs that are destroyed by finite $\mu$ in the Hermitian limit.
Phase Transitions: In the Rashba model, nonreciprocity renormalizes the chemical potential, shifting the topological phase transition boundary. The critical Zeeman field required for the topological phase increases with $\Gamma$ , but the non-Hermitian regime offers a wider parameter space for stable MZMs.
SN Junctions: In junctions where superconducting regions contact normal metals, the nonlocal MP effectively identifies accidental TZESs confined in the normal region (where $P^{LR} \approx 0$ ) versus true MZMs at the superconducting edges (where $P^{LR} \approx -1$ ).

Significance and Claims
The paper claims that the nonlocal biorthogonal Majorana polarization is a robust and necessary diagnostic tool for non-Hermitian topological superconductors. Its significance lies in:

Reliability: It provides a real-space indicator that remains effective even when the bulk-boundary correspondence is broken by non-Hermiticity.
Discrimination: It uniquely separates MZMs from EPs and TZESs, which is critical as EPs can mimic zero-energy states but lack topological protection.
Restoration of Topology: The work demonstrates that non-Hermitian effects are not merely perturbations but can actively enhance the stability and localization of MZMs, particularly in minimal chains where chemical potential usually suppresses topological features.
Experimental Relevance: The findings suggest that non-Hermitian effects, which naturally arise in experimental setups due to coupling with measurement probes or reservoirs, could be leveraged to improve the quality of Majorana modes in minimal Kitaev chain realizations.

The authors conclude that their framework offers a practical method to characterize zero-energy states in NH systems, providing a foundation for identifying true topological phases in the presence of gain, loss, and nonreciprocity.

Technical Summary: Nonlocal Majorana Polarization in Non-Hermitian Topological Superconductors

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