Robust quantized thermal conductance of Majorana floating edge bands in d-wave superconductors

This paper proposes and characterizes a new class of floating Majorana edge bands (FMEBs) in time-reversal symmetry-breaking $d$-wave superconductors, demonstrating through theoretical modeling and simulations that these isolated counterpropagating modes exhibit robust, quantized thermal conductance signatures that distinguish them from conventional chiral phases and offer a viable route for helical-like Majorana transport in topological quantum computation.

The Gist

Imagine you are trying to build a super-fast, unbreakable highway for tiny particles called electrons. In the world of quantum physics, there's a special type of particle called a Majorana fermion. Think of these as "ghost particles" because they are their own anti-particles. If you could get them to travel in a specific way, they could be the key to building super-powerful, error-proof quantum computers.

For a long time, scientists knew two main ways to build these highways:

1. The One-Way Street (Chiral): Like a highway where traffic only flows in one direction. If a car tries to go the wrong way, it hits a wall and stops. This is very stable but hard to control.
2. The Two-Way Street with a Guard (Helical): Like a highway where cars go both ways, but there's a magical force field (Time-Reversal Symmetry) that prevents them from crashing into each other. If you remove the force field, the highway collapses.

The Big Discovery
This paper introduces a brand-new, third type of highway called "Floating Majorana Edge Bands" (FMEBs).

Here is the simple breakdown of what the authors found, using some creative analogies:

1. The "Floating" Highway

Usually, edge states (the highways on the edge of a material) are glued to the main bulk of the material. They are like a road built right next to a massive mountain; you can't separate them.

The authors discovered a way to build a road that floats. Imagine a bridge suspended in mid-air, completely detached from the ground below. Even though the "ground" (the bulk material) is empty and has no special properties, this floating bridge exists perfectly in the gap. It's an isolated lane that doesn't touch the messy traffic below.

2. The "Ghost" Traffic Jam

In a normal two-way street, if a car going North meets a car going South, they might crash (scatter) and stop moving forward. This is bad for a quantum computer.

In this new "Floating" highway, the two lanes of traffic (one going North, one going South) are separated by a magical force. They are like two trains on parallel tracks that are so far apart they can never see each other. Even though they are on the same edge, they are "momentum-separated." Because they can't see each other, they can't crash. This makes the highway incredibly robust, even if the road is bumpy or dirty.

3. How They Built It: The "D-Wave" Recipe

How do you build this floating bridge? The authors used a special recipe involving a Quantum Anomalous Hall (QAH) insulator (a material that forces electrons to move in circles) and a d-wave superconductor (a special type of superconductor, like those found in high-temperature ceramics).

Think of the superconductor as a special paint. When you paint the QAH insulator with this "d-wave" paint, it doesn't just coat it evenly. It stretches the material in one direction and squishes it in the other (this is called anisotropy).

• This stretching and squishing acts like a sculptor chiseling away the "ground" (the bulk material) until the "bridge" (the edge states) is left floating in the air, completely detached.

4. The "Half-Quantum" Signature

How do we know this floating highway is real? The authors looked at how heat travels through it.

• Normal One-Way Highway: Heat flows perfectly in one direction.
• Normal Two-Way Highway: Heat flows both ways, but it's sensitive to noise.
• The Floating Highway: The heat flow is exactly half of what you'd expect from a normal particle.

Imagine a water pipe. If you have a full pipe, water flows at 100%. If you have a broken pipe, it flows at 0%. This new highway is like a pipe that is perfectly cut in half, and water flows at exactly 50% in both directions simultaneously. This "half-quantized" heat flow is the unique fingerprint that proves the floating highway exists.

5. Why It Matters

The best part? This highway is tough.

• Temperature: It works even if it gets a little warm.
• Disorder: It works even if the road is bumpy or has potholes (disorder).
• Chemical Potential: It works even if you tweak the voltage slightly.

The Bottom Line:
This paper says, "Hey, we found a new way to make these ghost-particle highways that doesn't need the usual 'magic force fields' (time-reversal symmetry) to stay stable. We can build them using materials we already know how to make (like magnetic topological insulators and cuprate superconductors). This opens a new door for building the quantum computers of the future."

It's like discovering a new type of bridge that floats in the air, is immune to wind and rain, and can carry a special kind of cargo that was previously thought impossible to transport.

Technical Summary

1. Problem Statement

Topological superconductors are predicted to host Majorana quasiparticles, which are crucial for fault-tolerant quantum computation. While conventional chiral Majorana edge modes (in time-reversal symmetry-breaking systems) and helical Majorana modes (in time-reversal invariant systems) are well-studied, a specific class of boundary states known as Floating Edge Bands (FEBs) has recently been identified in electronic and photonic systems. FEBs are distinct because they are isolated bands detached from the bulk continuum, extending across the entire Brillouin zone.

The Open Questions:

• Does an intrinsic superconducting analogue of FEBs exist?
• How can such a state emerge microscopically in a system that breaks time-reversal symmetry (TRS)?
• What are the unique transport signatures that distinguish these "Floating Majorana Edge Bands" (FMEBs) from conventional chiral or helical edge modes?

2. Methodology

The authors employed a multi-faceted theoretical approach combining analytical modeling and numerical simulations:

• Minimal Lattice Model: They constructed a two-band Bogoliubov–de Gennes (BdG) Hamiltonian on a square lattice. They introduced anisotropic Wilson mass terms ($B_x \neq B_y$) to investigate how lattice anisotropy affects the topological phase transition from a chiral superconductor ($C=1$) to a gapless edge phase ($C=0$).
• Microscopic Realization: They proposed a specific physical platform: a Quantum Anomalous Hall (QAH) insulator proximitized by a $d$-wave superconductor. They demonstrated that the $d$-wave pairing form factor ($\Delta(\mathbf{k}) \propto \cos k_x - \cos k_y$) naturally induces the required anisotropy in the Wilson mass terms when the system is transformed into the Majorana representation.
• Transport Simulations: Using the Nonequilibrium Green's Function (NEGF) method, they simulated electron and hole transport in both two-terminal and four-terminal geometries.
• Robustness Analysis: They systematically tested the stability of the predicted phenomena against:
  • Long-range correlated disorder.
  • Finite temperatures.
  • Finite chemical potential shifts (doping).
• Topological Diagnosis: They utilized winding numbers ($W_0, W_\pi$) defined on quasi-1D subsystems at high-symmetry momenta to distinguish the FMEB phase from trivial $N=0$ phases, as the bulk Chern number vanishes in both cases.

3. Key Contributions

• Proposal of FMEBs: The paper introduces Floating Majorana Edge Bands (FMEBs), a new class of boundary states in 2D superconductors that break TRS but host helical-like transport. These bands are momentum-separated, counterpropagating Majorana modes that are detached from the bulk continuum.
• Microscopic Mechanism: The authors identify Wilson mass anisotropy as the driving mechanism. Crucially, they show that coupling a QAH insulator to a $d$-wave superconductor naturally generates this anisotropy because the $d$-wave pairing term renormalizes the effective mass of the two decoupled Majorana blocks in opposite directions.
• Distinctive Transport Signatures: The study establishes a clear experimental fingerprint for FMEBs:
  • Two-terminal: Total thermal conductance remains quantized (unity), similar to chiral phases, making it insufficient for distinction.
  • Four-terminal (Single-edge): A robust half-quantized thermal plateau ($\kappa/T = \pi^2 k_B^2 / 6h$) appears in both propagation directions. This contrasts sharply with the chiral QAH phase (quantized in one direction, zero in the other) and the TRS helical phase (which is fragile to TRS-breaking disorder).

4. Key Results

• Phase Transition: Increasing the $d$-wave pairing strength ($\Delta$) drives the system from a chiral QAH phase ($N = \pm 2$) into an FMEB phase ($N=0$). In the FMEB phase, the single chiral edge mode reconstructs into a pair of counterpropagating Majorana modes separated in momentum space.
• Thermal Conductance:
  • In the FMEB regime, the thermal conductance along a single edge is half-quantized and bidirectional ($\kappa/T \approx 0.5$ in units of $\pi^2 k_B^2/3h$).
  • This half-quantization arises because each Majorana mode contributes half a fermionic channel, and the two counterpropagating modes allow transport in both directions.
• Robustness:
  • Disorder: FMEBs are robust against long-range disorder (due to momentum separation preventing backscattering) but are sensitive to short-range Anderson disorder. This distinguishes them from TRS-protected helical Majoranas, which are destroyed by TRS-breaking long-range disorder.
  • Temperature: The quantization remains stable up to moderate temperatures ($k_B T \sim 0.1$) due to a wide, flat transmission plateau near zero energy.
  • Chemical Potential: The FMEB phase remains stable under moderate chemical potential shifts, preserving the edge localization and zero-energy transmission.
• Topological Characterization: The FMEB phase is identified as a "weak topological" phase. While the total Chern number is zero, the winding numbers of the quasi-1D subsystems at high-symmetry cuts are non-trivial ($W = \pm 1$), confirming the topological nature of the floating bands.

5. Significance

• New Route to Helical Transport: The work demonstrates that helical-like Majorana transport can be achieved in systems without time-reversal symmetry, expanding the landscape of topological phases beyond the standard DIII class.
• Experimental Feasibility: The proposed platform (QAH insulator + $d$-wave superconductor) is experimentally accessible. Candidate materials include magnetically doped topological insulators (e.g., $(Bi,Sb)_2Te_3$, $MnBi_2Te_4$) interfaced with cuprate superconductors.
• Diagnostic Tool: The half-quantized, bidirectional thermal conductance provides a definitive experimental signature to distinguish FMEBs from conventional chiral Majorana modes and trivial phases, resolving ambiguities in previous QAH-superconductor experiments.
• Quantum Computing: By establishing a robust, anisotropy-stabilized Majorana transport channel, FMEBs offer a promising platform for exploring non-Abelian braiding and fault-tolerant quantum computation in solid-state systems.