🔬 mesoscale physics

Quantum theory for edge current and noise in two-dimensional topological superconductors

This paper theoretically demonstrates that while edge currents in two-dimensional topological superconductors vanish for non-chiral states, edge noise remains non-zero and serves as a topological indicator linked to the Chern number or other invariants like the Zak phase, with bulk noise acting as a topological susceptibility that peaks during phase transitions.

Original authors: S. Pintus, A. Crépieux

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: S. Pintus, A. Crépieux

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 two-dimensional superconductor not as a flat sheet of metal, but as a vast, invisible highway system for electrons. In a "topological" superconductor, this highway has a special rule: the electrons are forced to drive only on the very edge of the road, like cars stuck in a single lane along the shoulder, while the middle of the road (the "bulk") is completely empty and closed off.

This paper, written by S. Pintus and A. Cr´epieux, acts like a detective report investigating two things happening on this highway:

The Traffic Flow: How much current (electrons) is actually moving along the edge.
The Traffic Noise: The random jitters, bumps, and fluctuations in that flow, even when the traffic looks steady.

Here is what they discovered, explained through everyday analogies:

1. The Difference Between a One-Way Street and a Two-Way Street

The researchers looked at two types of "edge states" (the lanes where electrons travel):

Chiral (One-way streets): Electrons can only go in one direction.
Non-chiral (Two-way streets): Electrons can go forward or backward.

The Finding:

Current (The Flow): If the lane is a two-way street (non-chiral), the net traffic flow is zero. For every electron going left, another goes right, canceling each other out. But if it's a one-way street (chiral), there is a steady, non-zero flow of current.
Noise (The Jitters): This is the surprise. Even on a two-way street where the net flow is zero, there is still noise. It's like a busy intersection where cars are canceling each other out in terms of total movement, but the engine revs, honks, and tire screeches (fluctuations) are still loud and measurable. The paper proves that noise exists regardless of whether the edge is one-way or two-way.

2. The "Topological Susceptibility" (The Bulk Noise)

Usually, scientists look at the edge to find clues about the material's special properties. But this paper found something fascinating happening in the middle of the material (the bulk), where no current flows.

They discovered that the "noise" in the middle of the material acts like a seismograph for the material's shape.

The Chern Number: Think of this as a "topological ID card" or a "twist count" for the material. It tells you how many times the electron waves are twisted.
The Discovery: Every time the material changes its "twist count" (a topological phase transition), the noise in the middle of the material spikes dramatically.
The Metaphor: Imagine the material is a rubber band. If you twist it once, it's stable. If you twist it twice, it's stable. But the exact moment you are twisting it from one twist to two, the rubber band snaps and vibrates violently. The paper suggests that measuring the noise in the center of the material is a way to detect these "snapping" moments. They call this "bulk noise" a topological susceptibility—a way to measure how sensitive the material is to changing its fundamental shape.

3. The "Toy Models" They Tested

To prove this, they used two simplified mathematical models (like building a scale model of a bridge to test its strength):

The Qi-Wu-Zhang (QWZ) Model: A simple, single-layer highway.
The Bilayer Model: A highway with two stacked layers.

In both cases, they found the same pattern:

Edge Noise: Only appears when the "twist count" (Chern number) is non-zero. If the twist count is zero, the edge is quiet.
Bulk Noise: Peaks exactly when the twist count changes. It's like a warning bell that rings every time the material undergoes a topological transformation.

4. The Exception: When the Rules Change

The paper also looked at a more complex, "extended" version of the model (the extended Qi-Wu-Zhang model). In this specific case, the material had a "twist count" of zero (which usually means no edge states), yet it still had an edge state.

In this weird scenario, the usual rules broke down. The edge noise didn't follow the "twist count" anymore. Instead, it seemed to be listening to a different kind of topological rule called the Zak phase. This suggests that while the "twist count" is usually the boss, there are other hidden topological bosses that can take over in specific situations.

Summary

In simple terms, the authors developed a new mathematical toolkit to listen to the "static" (noise) of electrons in these special superconductors. They found that:

Edge Current only flows if the electrons are forced to go one way.
Edge Noise is always there, acting as a constant hum of activity.
Bulk Noise is the most interesting part: it acts like a topological sensor, screaming (spiking) every time the material's fundamental "twist" changes, even though no current is flowing in the middle.

This work provides a unified way to understand how these exotic materials behave, suggesting that listening to the noise might be just as important as measuring the current for understanding the quantum world.

Technical Summary: Quantum Theory for Edge Current and Noise in Two-Dimensional Topological Superconductors

Problem Statement
While topological superconductors are known for exotic properties such as topological phase transitions without symmetry breaking and the existence of chiral edge states hosting Majorana modes, theoretical understanding of edge current and noise in two-dimensional (2D) systems remains limited. Existing literature largely relies on quasi-classical approaches (e.g., Ginzburg-Landau theory, Eilenberger equations) or focuses on 1D wires and junctions. A unified quantum theory that explicitly accounts for the crystal lattice structure of 2D samples to calculate both edge current and its fluctuations (noise) has been lacking, despite the emergence of experimental techniques capable of measuring differential resistance and imaging edge currents.

Methodology
The authors develop a quantum theory based on T-matrix and Green function techniques to calculate the edge spectral function, edge current, and edge noise in a semi-infinite 2D system.

Modeling: The system is modeled as a square lattice with an edge at position $y=a$ , simulated by a strong impurity line at $y=0$ .
Green Functions: The retarded Green function for the semi-infinite plane is derived using the T-matrix formalism, relating the edge properties to the bulk Green function of the clean infinite system.
Current Calculation: The edge current is derived from the expectation value of the current operator between nearest neighbors. It is expressed in terms of the non-equilibrium Keldysh Green function ( $G^<$ ) and, in equilibrium, relates to the spectral function ( $A$ ) and the Fermi-Dirac distribution.
Noise Calculation: Finite-frequency noise is defined as the Fourier transform of the current-current correlator. Using Wick's theorem, the noise is expressed as an integral involving the product of lesser ( $G^<$ ) and greater ( $G^>$ ) Green functions. In equilibrium, this simplifies to an integral over energy and momentum involving the spectral function.
Application: The formalism is applied to two specific models with chiral edge states: the Qi-Wu-Zhang (QWZ) model and a bilayer model. The Chern number is calculated independently to correlate with the transport properties.

Key Contributions and Results

Distinction Between Current and Noise:
- Edge Current: The theory demonstrates that the edge current is non-zero only for chiral edge states. For non-chiral edge states, the current vanishes due to the presence of a $\sin(k_x a)$ factor in the current expression, which integrates to zero if the spectral function is even in $k_x$ .
- Edge Noise: In contrast, the edge noise is non-zero for both chiral and non-chiral edge states. This is because the noise expression contains a $\sin^2(k_x a)$ factor, which is always non-negative and does not vanish upon integration even for non-chiral states.
Relationship to Topological Invariants (Chern Number):
- Edge Noise: In models with non-zero Chern numbers ( $C \neq 0$ ), the edge noise is non-zero. When $C=0$ , the edge noise vanishes (in the absence of other topological invariants).
- Bulk Noise: The bulk noise is non-zero even when the bulk current is zero. Crucially, the bulk noise exhibits distinct peaks at chemical potential values ( $\mu$ ) where the Chern number changes. The authors suggest the bulk noise acts as a topological susceptibility, with peak heights proportional to the derivative of the Chern number with respect to $\mu$ .
- Edge Current Derivative: The derivative of the edge current with respect to $\mu$ ( $|d\langle I_a \rangle / d\mu|$ ) is found to roughly follow the dependence of the Chern number on $\mu$ , particularly when the superconducting order parameter is small relative to the hopping integral.
Model-Specific Findings:
- Qi-Wu-Zhang (QWZ) Model: Numerical results confirm that edge noise is non-zero only when $C \neq 0$ . Bulk noise peaks align with topological phase transitions (gap closings) where $C$ changes.
- Bilayer Model: Similar behaviors are observed, with edge noise tracking the non-zero Chern number intervals and bulk noise peaking at transition points. The edge current derivative exhibits a staircase behavior mirroring the Chern number steps.
- Extended Models (Zak Phase): The paper notes that in extended models where edge states exist despite a zero Chern number (e.g., due to other topological invariants like the Zak phase), the edge noise behavior deviates from the standard Chern-number dependence, linking instead to these other invariants.

Significance and Claims
The paper claims to provide the first unified quantum calculation of edge current and noise in 2D topological superconductors that explicitly incorporates the lattice structure. The primary significance lies in establishing noise as a probe for topological properties:

Bulk Noise as a Susceptibility: The authors propose that bulk noise can be viewed as a "topological susceptibility," providing a signature of topological phase transitions through peak fluctuations in current noise, even when the bulk current is zero.
Chirality Detection: The theory clarifies that while current is a strict indicator of chirality, noise is a more robust quantity that persists even in non-chiral edge states, though its magnitude and behavior are still tied to topological invariants in chiral systems.
Experimental Relevance: The results are framed as a theoretical foundation for interpreting emerging experimental techniques capable of measuring differential resistance and edge currents in Weyl superconductors and imaging edge currents.

The authors remain modest regarding the universality of these results, noting that future work is required to test these findings in more realistic systems and with different pairing symmetries (e.g., $s+id$). They explicitly state that the observed relationships between noise and topological invariants are specific to the models studied and may require modification in systems with different symmetries or in the presence of other topological invariants.