Quantifying robustness and locality of Majorana bound states in interacting systems

This paper rigorously establishes the connections between Majorana bound state separation, robust energy degeneracy, and protected non-abelian braiding in interacting systems by defining Majorana bound states from many-body ground states and demonstrating how their locality constrains environmental coupling to quantify protection.

The Gist

The Big Picture: Protecting the "Quantum Coin"

Imagine you are trying to build a super-advanced computer that uses the weird rules of quantum physics. A major problem is that these computers are incredibly fragile. A tiny bump, a stray magnetic field, or even a warm breeze can ruin the calculation.

To solve this, scientists look for a special kind of "quantum coin" called a Majorana Bound State (MBS). Think of an MBS not as a single particle, but as a pair of ghostly halves of a coin. One half lives at the very left end of a wire, and the other half lives at the very right end.

The Magic Trick:
If these two halves are far apart, they are "protected." If you poke the left side of the wire, you can't affect the right side. Because the information is split between two distant locations, local noise (like a bump in the middle of the wire) can't destroy the quantum state. This is called topological protection.

The Problem: When Things Get Messy (Interactions)

For a long time, scientists understood how to protect these coins if the particles inside the wire didn't talk to each other (non-interacting). But in real life, particles do talk to each other; they push, pull, and interact. This is called an interacting system.

When particles interact, the "ghostly halves" of the coin get messy. They aren't just simple points at the ends anymore; they become complex, fuzzy clouds that might stretch across the whole wire.

The Question:
In these messy, interacting systems, how do we know if the coin is still safe? How far apart are the halves really? And can we still do the "magic trick" of braiding them (swapping them around) to do calculations?

The Solution: A New Way to Measure "Distance"

The authors of this paper developed a new mathematical ruler to measure how "local" (how far apart) these messy Majorana halves really are, even when particles are interacting.

They used a concept called the Partial Trace.

• The Analogy: Imagine you have a giant, complex soup (the whole quantum system). You want to know how much "salt" (the Majorana particle) is in just the spoonful you are holding (a small region of the wire).
• The Method: Instead of looking at the whole soup, they mathematically "drain" everything outside the spoon. What's left in the spoon tells them how much of the Majorana particle is actually there.

If the spoon has almost no salt, the particle is far away. If the spoon is full of salt, the particle is right there.

What They Found

Using this new ruler, the authors proved three main things:

1. The "Safety Zone" is Quantifiable
They showed that if the "salt" (the Majorana particle) is very weak in the middle of the wire, the energy of the system is safe. It's like saying, "If the ghostly halves are truly separated, a local noise can't shake the coin." They created a formula that puts a hard limit on how much the energy can wiggle based on how well-separated the particles are.

2. The "Gauge" Problem (Choosing the Right Lens)
Because these particles are quantum, their appearance depends on how you look at them (a concept called "gauge"). The authors showed that you can "tune your glasses" to find the best view where the particles look most separated. They defined a Quality Score (like a grade for a student) that tells you how good your setup is.

• High Score: The particles are well-separated; the system is robust.
• Low Score: The particles are overlapping; the system is fragile.

3. Testing with Real Experiments
They tested their theory on a specific setup: a chain of quantum dots (tiny traps for electrons) that act like a wire.

• Disorder: They simulated "dirty" wires with random bumps. Their math predicted exactly how much the energy would split, and it matched computer simulations perfectly.
• Connecting to a Dot: They simulated connecting the wire to an extra quantum dot (an external sensor). They showed that if the Majorana particles are well-separated, the sensor won't disturb the system. If they are messy, the sensor will cause the energy to split, ruining the protection.

The "Braiding" Test

To do quantum computing, you have to move these particles around each other (braiding).

• The Analogy: Imagine trying to braid two ropes. If the ropes are stiff and far apart, it's easy. If they are tangled and mushy, it's a mess.
• The Result: The authors showed that their "Quality Score" predicts whether braiding will work. If the score is high (particles are local), you can swap them without errors. If the score is low, the swap will fail because the particles are too mixed up.

Summary

This paper doesn't invent a new machine; it invents a new ruler.

Before, scientists had to guess if their quantum systems were safe when particles were interacting. Now, they have a rigorous way to measure the "locality" of these particles. They can calculate a number that tells them:

1. How much the energy is protected from noise.
2. How likely it is that they can successfully perform quantum operations (braiding).

This is crucial for the next generation of quantum computers, which will likely rely on these messy, interacting systems rather than the simple, idealized ones of the past. It gives engineers a way to check their work and know if their "quantum coins" are truly safe.

Technical Summary

Technical Summary: Quantifying Robustness and Locality of Majorana Bound States in Interacting Systems

Problem Statement
Topological superconductors hosting separated Majorana bound states (MBSs) offer a mechanism for protecting qubits against local perturbations, a prerequisite for fault-tolerant quantum computing. In non-interacting systems, the spatial separation of single-particle MBSs rigorously implies robust ground-state degeneracy and the feasibility of non-abelian braiding. However, recent experimental progress in short quantum-dot-based Kitaev chains has highlighted systems where interactions are strong and MBS separation must be finely tuned. In these interacting regimes, the standard single-particle picture breaks down. It remains unclear how to rigorously quantify the locality of many-body MBS operators, how this locality constrains coupling to an environment, and under what conditions interacting systems retain protected degeneracy and non-abelian statistics.

Methodology
The authors develop a formalism applicable to both interacting and non-interacting systems by defining MBSs directly from the many-body ground states rather than single-particle wavefunctions.

1. Ground-State MBS Definition: For a fermionic system $S$ with two low-energy ground states of opposite fermion parity, $|e\rangle$ and $|o\rangle$, separated by a gap from excited states, the authors define ground-state MBS operators $\gamma$ and $\tilde{\gamma}$ acting within this subspace. These operators are analogous to Pauli matrices but are fermionic.
2. Partial Trace and Locality: To quantify the spatial localization of these non-local operators, the authors utilize the fermionic partial trace. The reduced MBS in a region $R$, denoted $\gamma_R = \text{Tr}_{\bar{R}}[\gamma]$, captures the local component of the operator.
3. Coupling Bounds: By coupling the system $S$ to an environment $B$ via a local interaction $H_{RB}$ acting in region $R$, the authors derive rigorous bounds on the effective coupling Hamiltonian in the ground-state sector. Using Schatten norms ($\|\cdot\|_p$), they establish inequalities relating the norm of the effective coupling operators to the norms of the reduced MBSs ($\|\gamma_R\|_q$) and the coupling strength.
4. Energy Splitting and Braiding: These coupling bounds are translated into bounds on the energy splitting ($\delta E$) between the ground states. Furthermore, the framework is extended to coupling-based braiding protocols involving multiple systems, deriving bounds on unwanted couplings that could spoil non-abelian statistics.

Key Contributions and Results

• Generalized Quality Measures: The paper introduces "quality factors" $Q_o$ and $Q_e$ that quantify the protection against odd and even parity perturbations, respectively. These are derived from the norms of reduced MBSs and local parity operators.
  • $Q_o = \sqrt{\sum_n \|\gamma_{R_n}\|_q^2}$ bounds the coupling to odd-parity operators.
  • $Q_e = \sqrt{\sum_n \|(i\gamma\tilde{\gamma})_{R_n}\|_q^2}$ bounds the coupling to even-parity operators.
  • The authors show that these measures depend on the gauge choice (relative phase between $|e\rangle$ and $|o\rangle$) and provide an analytical method to optimize this phase to minimize $Q_o$, thereby maximizing protection.
• Relation to Majorana Polarization: The work demonstrates that the optimized quality measures generalize the concept of Majorana polarization ($M$) used in non-interacting systems. Specifically, $M$ is shown to be the ratio of a "Majorana density" to a "fermion density" in the reduced region. The authors clarify that while $M$ bounds energy splitting for odd couplings in interacting systems (under specific conditions), the even-parity protection requires the distinct measure $Q_e$.
• Rigorous Bounds on Energy Splitting: The authors derive an inequality (Eq. 14) bounding the energy splitting $|\delta E|$ as a function of the coupling strength and the locality measures ($Q_o, Q_e$). The bound scales with the square root of the dimension of the environment's relevant subspace.
• Application to Interacting Kitaev Chain:
  • At a "sweet spot" of the interacting Kitaev chain, the authors analytically demonstrate that ground-state MBSs are localized on the outermost sites, despite having support on the full system.
  • Numerical simulations show that as the system is detuned from the sweet spot or subjected to disorder, the quality measures degrade, and the energy splitting increases, consistent with the derived bounds.
  • The bounds are shown to hold for weak disorder but can be violated when the disorder strength approaches the excitation gap, where higher-order perturbative effects become significant.
• Application to Quantum-Dot-Based Kitaev Chains:
  • The framework is applied to a minimal quantum-dot-based Kitaev chain model relevant to recent experiments.
  • The authors show that standard experimental tuning procedures (minimizing sensitivity of energy splitting to dot level fluctuations) primarily constrain even-parity perturbations ($Q_e$) but leave odd-parity protection ($Q_o$) unconstrained.
  • They link $Q_o$ to non-local conductance signatures, demonstrating that non-local conductance is bounded by the product of odd quality measures, providing a transport probe for the quality of MBS separation in interacting systems.
• Braiding Protocols: The authors analyze coupling-based braiding of three interacting systems. They derive bounds showing that successful braiding requires the desired couplings to be tunable while unwanted couplings (which depend on the overlap of MBSs in the coupling regions) remain small. The quality measures provide a criterion for assessing whether a specific system configuration is suitable for such protocols.

Significance and Claims
The paper claims to provide a rigorous, many-body framework for quantifying the robustness and locality of MBSs in interacting systems, a regime where single-particle intuition fails. By defining MBSs from ground states and utilizing partial traces, the authors establish a direct link between the spatial localization of these operators and the stability of the energy spectrum against local perturbations.

The authors state that their results:

1. Generalize the concept of MBS separation to interacting systems, allowing for the assessment of topological protection without relying on single-particle wavefunctions.
2. Provide physically meaningful measures ($Q_o, Q_e, M$) that can be calculated using standard numerical techniques (e.g., DMRG) for large interacting systems.
3. Clarify the limitations of existing quality measures (like Majorana polarization) in interacting contexts, showing they are sufficient for bounding odd-parity couplings but require generalization for even-parity protection.
4. Offer a theoretical basis for interpreting experimental data from quantum-dot-based Kitaev chains, specifically linking tuning procedures and transport signatures to the underlying topological quality of the system.

The work does not propose new experimental setups but rather provides the theoretical tools to analyze and quantify the performance of existing and proposed platforms in the presence of strong interactions. The authors emphasize that their bounds are rigorous within the low-energy subspace and assume weak coupling to the environment, noting that violations may occur at strong coupling or near the excitation gap.