Dissipation induced Majarona $0$- and $π$-modes in a driven Rashba nanowire

This paper demonstrates that dissipation in a periodically driven Rashba nanowire coupled to an s-wave superconductor induces both topological and trivial edge-localized Majorana 0- and $\pi$-modes, significantly modifying the system's topological phase diagram and enabling the creation of topological phases in non-topological setups.

The Gist

Imagine you are trying to build a very special, invisible fortress at the edge of a long, thin wire. This fortress is made of "Majorana modes," which are like ghostly particles that are their own anti-particles. Physicists have been hunting for these for decades because they could be the building blocks for super-powerful, unbreakable computers.

Usually, to build this fortress, you need a very specific, delicate setup: a wire with a special kind of spin (Rashba spin-orbit coupling), a magnetic field, and a superconductor nearby. But there's a catch: in the real world, nothing is perfectly isolated. Everything is always "leaking" energy into its surroundings, a process called dissipation. Usually, scientists pretend this leakage doesn't exist to make the math easier, but in reality, it's always there.

This paper asks a bold question: What happens if we stop pretending the leakage isn't there? Can we actually use this "leakage" (dissipation) and a rhythmic "pushing" (periodic driving) to build our fortress, or even build new kinds of fortresses we didn't know existed?

Here is what the authors found, explained through simple analogies:

1. The Setup: A Wire on a Trampoline

Think of the nanowire as a long trampoline.

• The Drive: Instead of just sitting there, the trampoline is being pushed up and down in a specific, rhythmic pattern (a "three-step drive"). This is like a drummer hitting the trampoline in a specific beat.
• The Dissipation: Now, imagine the trampoline is slightly wet or has holes in it, so energy leaks out as it bounces. This is the "dissipation."
• The Goal: The researchers wanted to see if they could create stable "ghosts" (Majorana modes) at the very ends of this leaking, bouncing trampoline.

2. The Two Types of Ghosts

The team discovered that this setup creates four types of edge states (ghosts at the ends of the wire), but they fall into two very different categories:

Category A: The "Real" Fortresses (Topological Modes)

These are the Majorana 0-modes and Majorana $\pi$-modes.

• The 0-modes are like the standard ghosts physicists have been looking for.
• The $\pi$-modes are a special new type that only exist because the trampoline is being rhythmically pushed. They are like ghosts that only appear when the drumbeat hits a specific note.
• Why they are special: These ghosts are "topological." Imagine they are tied to the fabric of the trampoline itself. You can't get rid of them just by shaking the trampoline a little; they are protected by the global shape of the system.
• The Twist: The authors found that the "leakage" (dissipation) actually helps! It can create these topological ghosts even in situations where a non-leaking system would be empty. It's like the rain (dissipation) helping the flowers (ghosts) grow in soil where they usually wouldn't survive.

Category B: The "Fake" Fortresses (Trivial Modes)

The researchers also found Trivial 0-modes and Trivial $\pi$-modes.

• These look exactly like the real ghosts at the edges of the wire. They sit right there, looking the same.
• The Catch: They are "trivial." They aren't protected by the global shape of the trampoline. Instead, they are created by "Exceptional Points" (EPs).
• The Analogy: Imagine two dancers spinning on the trampoline. Usually, they spin at different speeds. But at a specific moment (the Exceptional Point), they suddenly lock arms and spin as one single unit. This "locking" creates a temporary ghost at the edge. If you change the rhythm slightly, they un-lock, and the ghost disappears. These are fragile and not topologically protected, but they are still real phenomena caused by the interplay of the drive and the leak.

3. The Map of the World (Phase Diagram)

The authors drew a map (a phase diagram) showing where these ghosts appear.

• They found that by adjusting the "leakiness" (dissipation strength), you can switch between having ghosts and not having them.
• Crucially, they showed that dissipation can create topological phases that simply do not exist in a perfect, closed system. It's as if the rain creates a new type of island that never existed when the sun was shining.

4. Are They Real? (Robustness)

The team tested if these ghosts would survive if the trampoline had some bumps or dirt (disorder).

• Result: Both the "Real" (Topological) and "Fake" (Trivial) ghosts were surprisingly tough. They stayed stuck to the edges even when the system was messy.

Summary

In simple terms, this paper shows that imperfection (dissipation) is not just a nuisance; it's a tool. By combining a rhythmic push with a controlled leak, scientists can:

1. Create the famous "Majorana 0-modes."
2. Create a new type of "Majorana $\pi$-mode" that only exists in driven systems.
3. Create "Trivial" modes that look like the real ones but are caused by a different mechanism (Exceptional Points).
4. Use the leakiness to unlock topological phases that are impossible to reach in a perfect, closed world.

The paper concludes that this "driven-dissipative" approach offers a new, flexible way to engineer these exotic quantum states, potentially making them easier to create in real-world experiments where perfect isolation is impossible.

Technical Summary

Technical Summary: Dissipation Induced Majorana 0- and $\pi$-modes in a Driven Rashba Nanowire

Problem Statement
The search for Majorana zero-modes (MZMs) and Majorana $\pi$-modes (MPMs) has traditionally focused on closed, Hermitian systems. While periodic driving (Floquet engineering) offers a mechanism to induce topological phases and MPMs in otherwise trivial setups, real-world systems are invariably coupled to an environment, introducing dissipation. Dissipation is often viewed as detrimental to topological protection; however, it also introduces non-Hermitian (NH) effects, such as exceptional points (EPs), which can fundamentally alter system dynamics. This work addresses the gap in understanding how dissipation affects a periodically driven Rashba nanowire coupled to an $s$-wave superconductor. Specifically, it investigates whether MZMs and MPMs can survive in a driven-dissipative regime and whether dissipation can induce topological phases or new boundary modes not present in the closed system.

Methodology
The authors model a Rashba nanowire with proximity-induced $s$-wave superconductivity and an in-plane magnetic field, subject to a three-step periodic drive protocol. The system is coupled to a Markovian environment via linear onsite jump operators representing particle loss.

1. Formalism: The dynamics are governed by a periodic Liouvillian operator derived from the Lindblad master equation.
2. Third-Quantization: To handle the open quantum system, the authors employ the third-quantization method. They map the fermionic operators to real Majorana operators and construct a Liouville-Fock space. This allows the Liouvillian to be represented as a matrix operator acting on a vector space of density matrix elements.
3. Floquet Liouvillian: For the driven system, a time-evolution operator is constructed, leading to a "Floquet damping matrix" ($X_F$). The spectral and topological properties of the system are determined solely by this matrix, which acts as a non-Hermitian analog to the Floquet Hamiltonian.
4. Topological Characterization: The system possesses a generalized chiral symmetry (pseudo-anti-Hermitian symmetry). Using this symmetry, the authors define winding numbers ($\nu_0$ and $\nu_\pi$) based on the bulk properties of the damping matrix to distinguish between topological and trivial phases.

Key Results

• Existence of Majorana Modes: The driven-dissipative system supports both Majorana zero-modes (MZMs) at quasienergy $E=0$ and Majorana $\pi$-modes (MPMs) at $E=\pm \pi$. These modes are edge-localized and carry non-zero bulk topological invariants (winding numbers). Unlike their Hermitian counterparts, these modes acquire a finite imaginary part in their quasienergy, indicating a finite lifetime due to dissipation.
• Emergence of Trivial Modes: A novel finding is the appearance of "trivial" zero-modes (TZMs) and $\pi$-modes (TPMs). Like the Majorana modes, these are edge-localized and appear at $E=0$ and $E=\pm \pi$. However, they are topologically trivial (zero winding number) and their emergence is not tied to bulk gap closing/reopening. Instead, their creation and destruction are governed by the formation and annihilation of second-order exceptional points (EPs) in the spectrum.
• Dissipation-Induced Topology: The phase diagrams in the magnetic field ($B$) versus dissipation strength ($\gamma$) plane reveal that dissipation can substantially modify the topological phase diagram. Crucially, dissipation can induce topological phases (non-zero winding numbers) in parameter regimes where the corresponding closed (Hermitian) system is trivial. For instance, MZMs can be generated with lower magnetic field strengths in the presence of dissipation compared to the static or closed driven cases.
• Robustness: Both the topological (MZMs/MPMs) and trivial (TZMs/TPMs) edge modes are shown to be robust against onsite disorder.
• Absence of Skin Effect: Due to the specific form of the chosen jump operator (onsite loss), the system is free from the non-Hermitian skin effect, ensuring that the bulk-boundary correspondence for topological invariants remains valid.

Significance and Claims
The paper claims to extend the understanding of driven-dissipative topological superconductors by demonstrating that:

1. Dissipation does not merely destroy topological phases but can actively engineer them, inducing MZMs and MPMs in regimes where they would otherwise be absent.
2. The interplay between periodic driving and dissipation creates a rich landscape of boundary modes, distinguishing between topologically protected Majorana modes and EP-induced trivial modes.
3. The system provides a platform where non-Hermitian effects (specifically EPs) play a constructive role in shaping the spectral properties and dynamics of edge states.

The authors suggest that these findings are relevant for experimental setups involving semiconductor nanowires (e.g., InSb/InAs) coupled to superconductors and metallic leads, where controlled dissipation and step-drive protocols could be implemented to observe these phenomena. The work concludes by noting that while the study is limited to 1D, the principles could extend to higher-order topological superconductors in higher dimensions.