🔬 mesoscale physics

Non-Hermitian topological superconductivity with symmetry-enriched spectral and eigenstate features

This paper investigates a one-dimensional non-Hermitian superconducting lattice with all internal symmetries, demonstrating how the interplay of pseudo-Hermiticity, sublattice symmetry, and dissipation stabilizes Majorana zero modes and complex spectral features while establishing a topological invariant to map phase diagrams and reveal unique eigenstate correlations.

Original authors: Chuo-Kai Chang, Kazuma Saito, Nobuyuki Okuma, Hsien-Chung Kao, Chen-Hsuan Hsu

Published 2026-02-19

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CC BY 4.0

Original authors: Chuo-Kai Chang, Kazuma Saito, Nobuyuki Okuma, Hsien-Chung Kao, Chen-Hsuan Hsu

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 conductor trying to orchestrate a symphony of tiny particles (electrons) moving through a one-dimensional tunnel. In the "normal" world of physics (called Hermitian physics), this orchestra is perfectly balanced: energy is conserved, and the music flows smoothly without any surprises.

But in this paper, the authors are exploring a strange, "non-Hermitian" world. Think of this as a tunnel where the walls are slightly leaky (energy is lost to the environment) and the floor is slippery in one direction but sticky in the other (particles move easier to the right than to the left). This is the world of Non-Hermitian Topological Superconductivity.

Here is the story of what they discovered, broken down into simple concepts:

1. The Setup: A Wobbly, Leaky Ladder

The authors built a theoretical model of a ladder-like structure made of atoms.

The Ladder: It has two sides (Sublattices A and B), like a real ladder.
The Slippery Floor: Electrons can hop between rungs, but they prefer hopping right over left. This is called nonreciprocal hopping.
The Leaky Walls: The ladder is losing energy (dissipation). Some parts leak more than others.
The Magic Glue: They added "superconductivity," which acts like a magical glue pairing up electrons so they move together without resistance.

2. The Big Problem: The "Skin Effect"

In this weird, leaky world, there's a phenomenon called the Non-Hermitian Skin Effect. Imagine a crowd of people trying to walk down a hallway. In a normal hallway, they spread out evenly. But in this "skin" world, the crowd gets pushed entirely to one end of the hallway, piling up against the wall. This messes up the usual rules of physics, making it hard to find special "protected" states.

3. The Solution: The Magnetic "Traffic Cop"

To fix this pile-up, the authors introduced a uniform magnetic field (like a gentle, steady wind blowing across the ladder).

What it does: This wind pushes the crowd back into an even distribution, suppressing the "skin effect."
The Result: Once the crowd is organized, something magical happens: Majorana Zero Modes appear.

4. The Treasure: Majorana Zero Modes

Think of Majorana Zero Modes as "ghosts" or "invisible guardians" that live only at the very ends of the ladder.

They are special because they are their own antiparticles (a particle that is also its own mirror image).
They are incredibly stable and are the "holy grail" for building future quantum computers because they can store information without it getting corrupted by noise.
The Catch: The authors found that these ghosts only appear if the "leakiness" (dissipation) is uniform (the same on both sides of the ladder). If the leakiness is staggered (one side leaks, the other doesn't), the ghosts vanish. It's like trying to balance a seesaw; if one side is too heavy, the whole thing tips over and the magic disappears.

5. The Musical Spectrum: Real vs. Imaginary Tunes

In this quantum world, the "energy" of the system isn't just a number; it's a complex number with a real part and an imaginary part.

Real Energy: Like a normal, steady note.
Imaginary Energy: Like a sound that is purely fading away or growing.
Complex Flat Bands: The authors found regions where the music stops changing entirely, creating a "flat" line of sound regardless of where you look. This is a very unusual state where the system gets "stuck" in a specific configuration.

6. The Map: Winding Numbers

To prove these ghosts are real and not just a fluke, the authors created a Topological Map (called a Winding Number).

Imagine drawing a path on a map. If your path loops around a mountain (the center of the energy map) once, you have a "winding number" of 1. If it doesn't loop, it's 0.
They showed that whenever the path loops around the mountain (winding number $\neq$ 0), the Majorana ghosts appear at the ends of the ladder. If the path doesn't loop, the ghosts disappear. This map acts as a guarantee: if the map says "loop," the treasure is there.

7. The Hidden Connection: Left vs. Right

In normal physics, looking at a particle from the left or the right gives you the same picture. But in this non-Hermitian world, the "Left" view and the "Right" view are different.

The authors discovered a beautiful symmetry: The "Right" view of a particle is actually the "Left" view of a hole (an empty spot), and vice versa. It's like a perfect dance where the partners swap roles but stay perfectly in sync, enforced by the symmetries of the system.

Why Does This Matter?

This paper is like a blueprint for building a new kind of quantum machine.

Robustness: It shows how to use "loss" (dissipation) and "asymmetry" (nonreciprocity) not as bugs, but as features to create stable quantum states.
Experimental Roadmap: It tells experimentalists (people who build these things in labs using cold atoms or light) exactly what ingredients they need: a specific mix of slippery floors, uniform leaks, and a magnetic wind to catch the elusive Majorana ghosts.
New Physics: It reveals that when you mix superconductivity with these weird non-Hermitian rules, you get a richer, more colorful world of physics than we ever imagined possible.

In short: The authors found a way to tame a chaotic, leaky quantum system using a magnetic field to reveal hidden, stable "ghosts" at the edges, which could be the key to building the unbreakable computers of the future.

1. Problem Statement

The paper addresses the complex interplay between non-Hermiticity, superconductivity, and internal symmetries in one-dimensional lattice systems. While topological superconductivity (hosting Majorana zero modes) and non-Hermitian physics (featuring the skin effect and exceptional points) have been studied individually or in limited combinations, there is a lack of understanding regarding systems that simultaneously realize all internal symmetries permitted in non-Hermitian systems.

Specifically, the authors investigate how the combination of:

Sublattice symmetry (SLS) and Chiral symmetry (CS),
Pseudo-Hermiticity (pH) and Particle-Hole Symmetry (PHS),
Non-reciprocal hopping and onsite dissipation,
$s$ -wave superconducting pairing

shapes the spectral properties, topological invariants, and the stability of Majorana zero modes (MZMs). A key open question is how dissipation engineering (uniform vs. staggered) affects the existence and stability of these topological states.

2. Methodology

The authors employ a theoretical framework based on a generalized non-Hermitian Su-Schrieffer-Heeger (SSH) model with superconducting pairing.

Model Hamiltonian: They construct a 1D lattice model (Eq. 1) featuring:
- Sublattices A and B with $s$ -wave singlet pairing ( $\Delta_0$ ).
- Non-reciprocal hopping: Spin-dependent asymmetric hopping strengths ( $t \pm \sigma g/4$ ) within and between unit cells.
- Onsite dissipation: Complex onsite potentials ( $\Gamma_a, \Gamma_b$ ) representing gain/loss.
- Perturbation: A uniform transverse magnetic field ( $\delta h_x$ ) is introduced to suppress the non-Hermitian skin effect and stabilize boundary modes.
Symmetry Classification: The system is analyzed within the Altland-Zirnbauer (AZ) and AZ $^\dagger$ classification schemes. The authors identify the system as belonging to the BDI class (real AZ) and BDI $^\dagger$ class (real AZ $^\dagger$ ), characterized by Time-Reversal Symmetry (TRS), Particle-Hole Symmetry (PHS), and their dagger counterparts, along with SLS and pH.
Spectral Analysis:
- Periodic Boundary Conditions (PBC): Analytical derivation of the energy spectrum to identify gap-closing conditions, exceptional points, and regimes of real, purely imaginary, or complex flat bands.
- Open Boundary Conditions (OBC): Numerical diagonalization to observe the skin effect, boundary localization, and the emergence of zero modes.
Topological Invariant: The authors construct a Hermitianized Hamiltonian ( $\tilde{H}$ ) to define a spectral winding number ( $W$ ) as a topological invariant. This allows them to map the topological phase diagram and correlate the invariant with the appearance of MZMs.
Eigenstate Analysis: They investigate the density profiles of left and right eigenstates, deriving symmetry-enforced relations between particle/hole and spin components.

3. Key Contributions

Comprehensive Symmetry Realization: The model is the first to incorporate all possible internal symmetries (TRS, PHS, TRS $^\dagger$ , PHS $^\dagger$ , CS, SLS, pH) permitted in non-Hermitian systems within a superconducting lattice.
Stabilization Mechanism for MZMs: The authors identify a crucial condition for the existence of Majorana zero modes: uniform onsite dissipation is essential for stabilizing MZMs, whereas purely staggered dissipation ( $\Gamma_a = -\Gamma_b$ ) destroys the topological superconductivity and eliminates the zero modes.
Novel Spectral Phenomena: The study reveals rich spectral features absent in Hermitian systems, including:
- Complex Flat Bands: Energy bands that collapse to isolated points in the complex plane.
- Mixed Real/Imaginary Spectra: Regimes where half the bands are real and the other half are purely imaginary.
- Gapless Superconductivity: Emergence of nodal points away from high-symmetry momenta due to the interplay of non-reciprocity and non-Hermiticity.
Symmetry-Enriched Eigenstate Correlations: The paper uncovers non-trivial correlations between the components of left and right eigenstates. Specifically, the particle component of a right eigenstate is identical to the hole component of the left eigenstate (with the same spin), a relation enforced by the combined action of Chiral Symmetry and Pseudo-Hermiticity.
Topological Phase Diagrams: By constructing the winding number via Hermitianization, the authors establish a direct correspondence between the topological invariant and the gap-closing conditions, mapping out phase diagrams for both uniform and non-uniform dissipation regimes.

4. Key Results

Gap Closing Conditions: The gap closes either at high-symmetry points ( $k=0, \pi$ ) forming hyperbolic curves in parameter space, or at generic momenta when specific relations between hopping and dissipation parameters are met (leading to exceptional points).
Role of Transverse Field: The uniform transverse magnetic field ( $\delta h_x$ ) is critical for suppressing the non-Hermitian skin effect, allowing the OBC spectrum to asymptotically approach the PBC spectrum and revealing the topological zero modes.
Dissipation Dependence:
- Uniform Dissipation ( $\Gamma_a = \Gamma_b$ ): Supports a robust topological phase with MZMs.
- Staggered Dissipation ( $\Gamma_a = -\Gamma_b$ ): The gap-closing curves shift such that the topological region (where winding number $\neq 0$ ) vanishes; no MZMs are observed.
Winding Number: The winding number $W$ takes values of $0, \pm 1$ . $W \neq 0$ corresponds to the presence of MZMs. The invariant changes by $\pm 1$ across standard gap-closing curves and by $\pm 2$ across the lines defining gapless superconducting phases.
Robustness: The topological phase and MZMs are robust against small staggered pairing components and small disorder in dissipation, provided the uniform component of dissipation remains dominant.

5. Significance

Theoretical Advancement: This work significantly expands the classification of non-Hermitian topological phases by demonstrating how the interplay of multiple symmetries (specifically the "dagger" symmetries) enriches the structure of eigenstates and spectra. It provides a rigorous framework for understanding how dissipation can be engineered to stabilize rather than destroy topological order.
Experimental Relevance: The proposed model is realizable in current experimental platforms, including:
- Ultracold atomic gases (using optical Feshbach resonances for pairing and controlled atom loss for dissipation).
- Optical waveguides (for gain/loss engineering).
- Electrical circuits (for tunable non-reciprocal parameters).
- Mechanical/Acoustic metamaterials.
Quantum Computing: By identifying the conditions for stabilizing Majorana zero modes in dissipative environments, the paper offers new pathways for realizing fault-tolerant qubits in open quantum systems, moving beyond the idealized Hermitian limit.
New Physics: The discovery of "symmetry-enriched" correlations between left and right eigenstates suggests new avenues for probing non-Hermitian quantum mechanics, potentially detectable via spin-resolved scanning tunneling microscopy (STM) in solid-state realizations.

In conclusion, the paper establishes that non-Hermiticity, when combined with sublattice structure and superconductivity, does not merely perturb topological phases but fundamentally enriches them, creating new spectral features and requiring specific dissipation engineering to protect topological zero modes.