⚛️ quantum physics

Bogoliubov quasi-particles in superconductors are integer-charged particles inapplicable for braiding quantum information

This paper presents a rigorous proof that under number-conserving Hamiltonians, Bogoliubov quasi-particles possess integer charges identical to bare particles, thereby demonstrating that Majorana zero modes cannot be their own antiparticles suitable for braiding-based quantum computation and necessitating a fundamental re-evaluation of the Majorana approach and the symmetry-breaking assumptions in superconductivity theory.

Original authors: Zhiyu Fan, Wei Ku

Published 2026-06-23

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CC BY 4.0

Original authors: Zhiyu Fan, Wei Ku

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

The Big Idea: A "Mathematical Reality Check"

Imagine you are trying to build a super-secure vault (a quantum computer) using a very specific type of lock (Majorana particles). For decades, scientists have believed these locks exist inside superconductors (materials that conduct electricity with zero resistance). They thought these locks were special "ghost" particles that were their own anti-particles, allowing them to be twisted around each other like braided hair to store information safely.

This paper argues that the locks don't exist.

The authors, Zhiyu Fan and Wei Ku, present a rigorous mathematical proof showing that the particles scientists have been studying (called Bogoliubov quasi-particles) are actually just normal, integer-charged electrons in disguise. Because they are normal electrons, they cannot be twisted into the special "braids" needed for this type of quantum computing.

The Core Conflict: The "Rule of Conservation"

To understand why, we need to look at a fundamental rule of the universe: Conservation of Number.

The Analogy: Imagine a bank vault where you can only move gold coins in and out. You can never create a coin out of thin air, and you can never make a coin disappear. If you have 10 coins, you must always have 10 coins.
The Problem: The current popular theory of superconductivity acts like a magician who claims they can turn a coin into a "half-coin" and a "negative half-coin" simultaneously. This is mathematically convenient, but it breaks the rule that you must always have whole coins.
The Paper's Claim: In real-world materials (condensed matter), electrons are like those gold coins. You cannot have "half" an electron or a mix of an electron and a "hole" (a missing electron) that acts as a single, new particle. The math proves that if you strictly follow the rule of "whole coins only," the "ghost particles" (Majorana modes) simply cannot exist.

The "Dressed" Electron vs. The "Ghost"

The paper distinguishes between two ways of looking at these particles:

The "Ghost" View (Current Theory): Scientists often describe these particles as a "coherent superposition" of an electron and a hole. Think of this like a chameleon that is simultaneously red and blue. Because it's a mix, the theory says it can be its own anti-particle (like a mirror image of itself).
The "Dressed" View (This Paper's Proof): The authors argue that in reality, these particles are just electrons wearing a heavy coat. They are "dressed" by interacting with other electrons, but underneath the coat, they are still just one single electron.
- The Metaphor: Imagine a person wearing a heavy winter coat, a hat, and a scarf. If you look from far away, they might look like a strange, bulky shape. But if you apply the "Law of Conservation" (counting the actual body parts), you realize: "That is still just one human." They are not a half-human/half-robot hybrid. They are a human with accessories.

Because these "dressed" electrons are still just one electron, they carry a full electric charge. They cannot be their own anti-particle. Therefore, they cannot perform the "braiding" trick required for the proposed quantum computer.

Why This Matters for Quantum Computing

The proposed "Majorana" route to quantum computing relies on the idea that these particles are "non-Abelian."

The Braiding Analogy: Imagine you have two strings. If you swap them around each other in a specific way, the knot changes in a way that remembers the order of the swap. This is how the proposed computer would store data.
The Paper's Verdict: Since the particles are actually just normal electrons (not the special "ghost" hybrids), swapping them around doesn't create a new knot. It just puts them back where they started. The "magic" knot doesn't exist.

The Challenge of "Making" the Magic

The paper also points out a second problem: Even if we wanted to force these special states to exist, it would be incredibly hard to do so physically.

The Analogy: Imagine a room full of people standing in a circle. If you want them to stand in a specific, complex pattern that requires them to hold hands in a way that defies their natural tendency to stand apart, you can't just ask them to "imagine" it. You have to physically move them.
The Reality: The paper argues that natural physical processes (like heating up a material or applying a slow magnetic field) will always force the system back into "normal" states (the product states of individual electrons). Creating the "artificial" superposition needed for the quantum computer would require breaking the fundamental rules of how electrons behave in nature, which is physically impossible under normal conditions.

The Conclusion: Back to the Drawing Board

The authors conclude that the popular theory of superconductivity, which relies on "spontaneously broken symmetry" (the idea that the system chooses a state that breaks the conservation rule), is a useful approximation for some things, but it is conceptually wrong when it comes to quantum information.

They are calling for a return to a "number-conserving" theory—a way of describing superconductors that respects the rule that you can't create or destroy electrons out of thin air. Until we have a theory that respects this rule, the quest for this specific type of quantum computer is built on a misunderstanding of what these particles actually are.

In short: The paper says, "We thought we found a magical particle that could braid information. We proved mathematically that it's just a normal electron in a coat. The magic trick doesn't work, and we need to find a new way to build our quantum computers."

Technical Summary: "Bogoliubov quasi-particles in superconductors are integer-charged particles inapplicable for non-Abelian quantum information braiding"

Problem Statement
The paper addresses a fundamental conceptual conflict in the theoretical description of superconductivity and its application to topological quantum computation. The prevailing route for fault-tolerant quantum computation relies on the existence of Majorana zero modes (MZMs) in the vortex cores of topological superconductors. These modes are theoretically justified as Bogoliubov quasi-particles that are their own antiparticles (self-conjugate), thereby exhibiting non-Abelian statistics suitable for braiding quantum information. However, this justification relies on the standard effective field theory description of superconductivity, which assumes a spontaneously broken $U(1)$ symmetry and allows for particle number fluctuations. The authors argue that this description violates the strict particle number conservation inherent to the dynamics of electrons in condensed matter systems, where energy scales are too low to permit the creation or annihilation of bare particles. Consequently, the validity of treating Bogoliubov zero modes as Majorana particles and using them for non-Abelian braiding is questioned.

Methodology
The authors employ a rigorous mathematical framework based on exact $N$ -body quantum mechanics with a number-conserving Hamiltonian ( $H$ ).

Number Conservation Constraint: They establish that for any sensible description of electrons in condensed matter, the Hamiltonian must commute with the total particle number operator ( $[N_c, H] = 0$ ). This implies that eigenstates of $H$ must reside within subspaces of fixed particle number $N$ .
Eigen-Particle Formalism: The authors utilize the concept of "eigen-particles" ( $\tilde{c}^\dagger_l$ ), defined via a unitary transformation ( $U^\dagger$ ) of bare particle operators ( $c^\dagger_l$ ) within the $N$ -body Hilbert space. These eigen-particles are constructed to diagonalize the Hamiltonian while strictly preserving the particle number.
Comparison with Effective Field Theory: The study contrasts the rigorous eigen-particle description with the standard effective field theory (EFT) approach (e.g., BCS mean-field theory), which introduces approximate number-fluctuating terms (like $c^\dagger c^\dagger$ ) to generate a coherent superposition of particle and hole states.
Theorem Proving: The authors derive two key theorems based on the commutation relations of the number operator with the eigen-particle creation operators.

Key Contributions and Results

Quantized Charge and Mass of Quasi-particles:
The authors prove Theorem 1: Under a number-conserving Hamiltonian, one-body eigen-particles carry exactly one unit of bare particle number. Consequently, they possess quantized electric charge and inertial mass identical to the bare electrons. This result is derived from the commutation relation $[N_c, \tilde{c}^\dagger_l] = \tilde{c}^\dagger_l$ , which dictates that eigen-particles increase the particle number by exactly 1.
Impossibility of Majorana Nature:
Based on Theorem 1, the authors prove Theorem 2: Eigen-particles cannot be their own antiparticles (Majorana particles). If an eigen-particle were self-conjugate ( $\tilde{c}^\dagger_l = \tilde{c}_l$ ), the commutation relation would imply $\tilde{c}^\dagger_l = -\tilde{c}^\dagger_l$ , leading to the trivial solution $\tilde{c}^\dagger_l = 0$ . Therefore, proper Bogoliubov quasi-particles in a number-conserving system are massive, integer-charged fermions, not Majorana fermions.
Invalidation of Non-Abelian Braiding:
The study concludes that the standard identification of Bogoliubov zero modes as Majorana modes is an artifact of the number-non-conserving approximation. In a rigorous number-conserving treatment, the zero modes in vortex cores are merely charged electrons (or holes) and do not possess the self-conjugate property required for non-Abelian statistics.
Furthermore, the authors analyze the structure of degenerate ground states. They argue that in a number-conserving system, the degenerate subspace size scales as $\binom{L}{M}$ (combinatorial) rather than the $2^{L/2}$ scaling predicted by Majorana theories. A braiding operation on these states results in the original product state (Abelian phase factor of 1) rather than a transformation within a non-Abelian manifold.
Challenge in State Preparation:
The paper highlights a conceptual challenge in preparing "braidable" states. While one can mathematically construct extensive superpositions of degenerate product states (which might exhibit non-Abelian properties), such states require artificial coherence that is not generated by the physical thermalization dynamics of the number-conserving Hamiltonian. The authors argue that physical processes (thermalization or slow external fields) naturally drive the system into product states of eigen-particles, making the preparation of the required entangled superpositions physically unfeasible without explicitly breaking number conservation.

Significance and Claims
The paper claims to provide a rigorous proof that the current theoretical foundation for Majorana-based quantum computation is conceptually flawed when applied to systems governed by strict particle number conservation.

Correction of Misconception: It asserts that the widely accepted view of Bogoliubov quasi-particles as coherent superpositions of electrons and holes is an artifact of effective field theories that violate particle conservation.
Re-evaluation of Quantum Computing Routes: The authors state that the "heavily pursued" route of using Bogoliubov zero modes for non-Abelian braiding requires serious re-consideration, as these modes are not Majorana in nature within a proper many-body framework.
Call for New Theory: The study aims to reignite the search for a proper, number-conserving, $U(1)$ symmetric theory of superconductivity. It suggests that the general eigen-particle representation offers a convenient framework to encode essential many-body dynamics without introducing unphysical properties like artificial coherence or broken symmetry.
Scope of Impact: The authors emphasize that while effective field theories remain successful for describing one- and two-body properties (like spectral functions or tunneling) where particle conservation is not the primary concern, they fail fundamentally when applied to $N$ -body quantum information and topological braiding.

The paper does not propose new experimental protocols to fix this issue but rather suggests that achieving braidable states would likely require explicitly breaking number conservation (e.g., via strong coupling to an environment with electron exchange), a route that is currently under-explored.