Nonlocal Cooper pairs in finite topological superconductors and their relation to Majorana nonlocality

This paper identifies that finite one-dimensional topological superconductors host unconventional nonlocal Cooper pairs in the low-energy regime, where normal and anomalous Green's functions become identical and exhibit exponentially growing correlations between system ends, thereby linking these pair correlations to fermion parity and the nonlocality of hybridized Majorana modes.

The Gist

Imagine a superconductor not as a solid block of metal, but as a long, narrow hallway. In a normal hallway, if you shout from one end, the sound gets quieter the further it travels until it disappears. But in the special "topological" superconductors described in this paper, something magical happens at the very low energies (like a whisper).

Here is the story of what the researchers found, explained through simple analogies:

1. The Two Ways to Look at the Hallway

Scientists usually study superconductors in two ways:

• The "Single Particle" View: Looking at individual electrons running down the hallway.
• The "Pair" View: Looking at Cooper pairs (electrons that hold hands and dance together).

Usually, these two views tell different stories. However, the authors discovered that in these specific, finite-length superconductors, the two views become identical twins. At low energies, the behavior of a single electron and the behavior of a dancing pair are exactly the same, just wearing a slightly different "mask" (a phase factor). It's as if the electron and its partner are so deeply connected that you can't tell them apart anymore.

2. The Magic of "Nonlocality" (The Ghost Connection)

This is the paper's biggest discovery. In a normal system, if you look at the connection between the two ends of the hallway (the left wall and the right wall), it should be weak because they are far apart.

But in this topological superconductor, the connection between the two ends grows stronger the longer the hallway gets.

• The Analogy: Imagine two people standing at opposite ends of a very long bridge. In a normal bridge, they can't hear each other. But in this "topological" bridge, the longer the bridge gets, the louder they can hear each other. Their connection actually increases with distance.
• The Local Silence: Meanwhile, if you try to listen to what's happening right next to one person (local correlation), it goes completely silent. The "action" is entirely happening between the two far ends, ignoring the middle.

The researchers call these "unconventional nonlocal Cooper pairs." They are pairs of electrons that are linked across the entire length of the material, ignoring the space in between.

3. The "Majorana" Ghosts

Why does this happen? The paper explains that at the two ends of this hallway, there are special "ghosts" called Majorana modes.

• Think of these ghosts as half-electrons. One ghost lives at the left end, and its twin lives at the right end.
• Normally, these ghosts are stuck at their ends. But because the hallway is finite (it has a start and an end), these two ghosts can "shake hands" across the distance.
• When they shake hands, they form a single, invisible "nonlocal fermion" that exists everywhere at once. The "nonlocal Cooper pairs" the authors found are essentially the physical manifestation of these two ghosts holding hands across the gap.

4. Why This Matters (The "Qubit" Connection)

The paper links this strange behavior to Fermion Parity.

• Imagine a light switch that can be either "On" or "Off." In this system, the state of the entire system (whether the "ghost handshake" is active or not) acts like a single bit of information.
• Because this information is stored in the connection between the two far ends (and not in the middle), it is very hard to disturb. This is the core idea behind topological quantum computing: storing data in a way that is protected from noise.
• The authors show that the strange "nonlocal Cooper pairs" are directly responsible for how this information is stored and how electricity flows through the system in a unique way (specifically, how electrons can tunnel from one end to the other without getting stuck).

Summary

The paper reveals that in finite topological superconductors:

1. Single particles and pairs are twins: They behave identically at low energies.
2. Distance is an advantage: The connection between the two ends gets stronger as the system gets longer, while local connections vanish.
3. The "Ghost Handshake": This is caused by Majorana modes at the ends linking up, creating a special type of electron pair that spans the whole system.
4. The Big Picture: This behavior is the physical proof of "Majorana nonlocality," a key concept for building future quantum computers that are robust against errors.

The authors didn't just guess this; they used complex math (Green's functions) to prove it, and then ran computer simulations to confirm that these "ghostly" connections really do exist and behave exactly as the math predicts.

Technical Summary

1. Problem Statement

Topological superconductors (TSCs) hosting Majorana end modes are central to the field of topological quantum computation. While the quasiparticle picture (Bogoliubov–de Gennes formalism) has extensively characterized Majorana modes and their nonlocality (specifically the hybridization of end modes into a single nonlocal fermionic mode), the pair-correlation picture (Gor'kov Green's function formalism) has been less explored in this context.

Existing literature often relies on simplified low-energy effective Hamiltonians ($H_M = i \frac{E_M}{2} \gamma_a \gamma_b$) to explain transport phenomena like nonlocal conductance and noise. However, a rigorous analytical derivation of the full Gor'kov Green's function for finite one-dimensional TSCs is lacking. Specifically, it is unclear how the emergence of nonlocal fermionic modes manifests in terms of Cooper pair correlations (anomalous Green's functions) and how these relate to the equivalence between single-particle and pair correlations in the low-energy limit.

2. Methodology

The authors employ an analytical approach based on the Kitaev chain model, a prototypical 1D topological superconductor.

• Model: They utilize the Bogoliubov–de Gennes (BdG) Hamiltonian for a finite chain of length $L$ with hopping $t$, chemical potential $\mu$, and p-wave pairing $\Delta$.
• Derivation:
  1. They first derive the bulk Matsubara Green's function using the continuum limit approximation ($\Delta, \omega \ll t$).
  2. They introduce hard-wall boundary conditions (via infinite boundary potentials) to solve the Dyson equation for a finite system.
  3. They perform an asymptotic expansion in the low-frequency regime ($\omega \ll \Delta_{k_F}$) and for long systems ($L \gg \xi_0$, where $\xi_0$ is the coherence length).
• Analysis: The derived Green's functions are decomposed into normal ($G$, single-particle) and anomalous ($F$, Cooper pair) components. The authors analyze their spatial dependence, frequency dependence, and symmetry properties.
• Transport Calculation: To connect the Green's functions to observables, they calculate the scattering matrix for a multiterminal setup (TSC connected to two normal leads) using the wide-band limit approximation.
• Validation: Analytical results are cross-verified with numerical simulations using the recursive Green's function method.

3. Key Contributions

The paper establishes two fundamental properties of the Gor'kov Green's function in finite TSCs that were previously unexplored in the pair-correlation framework:

1. Equivalence of Normal and Anomalous Green's Functions: In the low-energy regime, the normal Green's function (describing electron correlations) and the anomalous Green's function (describing Cooper pair correlations) become identical up to a phase factor. This implies that electrons and holes are effectively equivalent at low energies, reflecting the Majorana nature of the system.
2. Pronounced Nonlocality: The Green's functions exhibit a unique spatial behavior:
   • Local correlations (at the edges, e.g., $G(1,1)$) vanish as $\omega \to 0$ (odd-frequency symmetry).
   • Nonlocal correlations (between opposite ends, e.g., $G(1,L)$) grow exponentially with the system length $L$ in the zero-frequency limit.

4. Key Results

A. Analytical Form of Green's Functions

For a finite chain of length $L$ in the low-frequency limit ($\omega \ll E_M$, where $E_M$ is the hybridization energy), the Green's functions at the ends ($j=1, j'=L$) are derived as:
$$\hat{G}(1, L) \propto \frac{E_M}{\omega^2 + E_M^2} (\hat{\tau}_1 - i\hat{\tau}_3)$$
$$\hat{G}(1, 1) \propto \frac{\omega}{\omega^2 + E_M^2} (1 - \hat{\tau}_2)$$

• Nonlocal Term: The amplitude scales as $E_M \propto e^{-L/\xi_0}$. However, the ratio of the nonlocal term to the local term leads to an effective growth of correlations with length in the zero-frequency limit ($\lim_{\omega \to 0} \hat{G}(1,L) \propto e^{L/\xi_0}$).
• Local Term: The local Green's function is proportional to $\omega$, vanishing at zero frequency (odd-frequency pairing).

B. Emergence of Nonlocal Cooper Pairs

The authors identify the existence of "unconventional nonlocal Cooper pairs." These are pairs where the two electrons are spatially separated at opposite ends of the wire. The magnitude of the anomalous Green's function $|F(1,L)|$ is equal to the normal Green's function $|G(1,L)|$, and both scale exponentially with $L$.

C. Connection to Fermion Parity

The nonlocal correlator $\langle c_1 c_L^\dagger \rangle$ is directly linked to the fermion parity operator ($P_f$) of the nonlocal fermionic mode formed by the hybridized Majorana modes:
$$\langle P_f \rangle \propto \sum_\omega G(1, L) \propto \sum_\omega F(1, L)$$
This establishes a direct mathematical link between the pair correlation function and the topological qubit state (parity).

D. Transport Signatures

The authors re-examine charge transport in a multiterminal setup. They demonstrate that the scattering coefficients derived from their Green's functions naturally reproduce the known signatures of Majorana nonlocality:

• Dominance of Inter-lead Scattering: Cross-lead scattering dominates over local Andreev reflection.
• Equal Probability: Elastic cotunneling (electron-electron) and crossed Andreev reflection (electron-hole) occur with equal probability.
• Noise Anomaly: These properties lead to a maximal positive cross-correlation in current noise ($2P_{ab} = P_{aa} + P_{bb}$), which is explained here as a direct manifestation of the nonlocal Cooper pairs.

5. Significance

• Bridging Frameworks: The work successfully bridges the quasiparticle picture (Majorana modes) and the pair-correlation picture (Cooper pairs). It proves that Majorana nonlocality is not just a single-particle phenomenon but is intrinsically encoded in the structure of Cooper pairs.
• New Perspective on Qubits: By linking the nonlocal Green's function directly to fermion parity, the paper provides a robust theoretical foundation for understanding how topological qubits (encoded in parity) can be probed via pair correlations.
• Experimental Relevance: The results offer a theoretical basis for interpreting multiterminal transport experiments and noise measurements as probes of nonlocal Cooper pairs, potentially aiding in the detection of Majorana modes in finite systems (e.g., semiconductor nanowires or magnetic atom chains).
• Beyond Effective Models: Unlike previous studies relying on simplified effective Hamiltonians, this work derives these properties from the full microscopic Hamiltonian, confirming their robustness against approximations.

In conclusion, the paper reveals that finite topological superconductors host a unique state of matter where single-particle and Cooper-pair correlations are indistinguishable at low energies and exhibit exponential nonlocality, providing a deeper understanding of the "Majorana nonlocality" essential for topological quantum computation.