Equivalent class of Emergent Single Weyl Fermion in 3d Topological States: gapless superconductors and superfluids Vs chiral fermions

This paper proposes a generic approach using spontaneous $U(1)$ symmetry breaking to construct 3D lattice models that evade the no-go theorem and yield a single Weyl fermion in the infrared limit, demonstrating that these models form an equivalent class with gapless superconductors and superfluids across three distinct symmetry-breaking pathways.

The Gist

The Big Picture: The "Impossible" Particle

Imagine you are trying to build a specific type of Lego tower. In the world of physics, there is a famous rule called the Nielsen-Ninomiya No-Go Theorem. It's like a cosmic building code that says: "You cannot build a single, isolated tower of this specific shape on a grid. If you build one, the universe forces you to build a mirror-image twin right next to it."

In physics terms, you can't have just one "Weyl fermion" (a special, massless particle that acts like a one-way street for electricity) on a 3D lattice. They always come in pairs: a left-handed one and a right-handed one.

The Goal of this Paper:
The authors, Gabriel Meyniel and Fei Zhou, want to break this rule. They want to build a lattice model (a digital grid) that hosts only one single Weyl fermion. They want to see how this "impossible" particle emerges from the chaos of a superconductor or superfluid.

The Secret Weapon: Breaking the Rules

How do they break the cosmic building code? They realize the code relies on a specific rule: Charge Conservation. The universe usually insists that electric charge is perfectly conserved (like money in a bank account that never changes).

The authors say: "What if we stop counting the money?"

They propose building their models using Superconductors or Superfluids. In these states, particles pair up and flow without resistance, effectively breaking the strict rule of charge conservation. By "breaking the bank," they can evade the No-Go theorem and isolate a single Weyl fermion.

The Three Paths to the Solution

The authors explore three different "construction sites" (paths) to achieve this single particle. Think of these as three different ways to sculpt a statue from a block of marble.

Path A: The "Critical Tipping Point" (The Seesaw)

• The Setup: Start with a stable, gapped material (a solid block of marble).
• The Action: You slowly tune the material until it hits a "critical point" where the energy gap closes. It's like balancing a seesaw perfectly in the middle.
• The Result: At this exact tipping point, the material becomes a Topological Quantum Critical Point (tQCP). Here, the "extra" particles cancel each other out, leaving behind a single, emergent Weyl fermion.
• Analogy: Imagine a crowded dance floor where everyone is paired up. You slowly change the music until, at one specific moment, all the couples separate except for one dancer who is now spinning alone in the center.

Path B: The "Peeling" Method (The Onion)

• The Setup: Start with the same stable block.
• The Action: Instead of waiting for a tipping point, you apply a strong magnetic field. This field acts like a peeler. It strips away the "excessive" degrees of freedom (the extra particles) that usually come in pairs.
• The Result: The magnetic field pushes the unwanted partners away, leaving behind a pair of "nodal points" (holes in the energy). Because of the special symmetry of these real fermions, these two points can be mathematically reinterpreted as a single Weyl fermion.
• Analogy: Imagine you have a double-decker bus full of passengers. You open the doors and the magnetic field forces the top deck to empty out. Now you only have the bottom deck. You then realize that the two seats on the bottom deck are actually just one "super-seat" viewed from a different angle.

Path C: The "Hybrid" Approach (The Remix)

• The Setup: This is a mix of the two above. You start with a complex critical point (Path A) that has too many particles, and then you apply the magnetic field (Path B) to "peel" away the excess.
• The Result: You can take a model that has 4 or 8 particles and use the magnetic field to whittle it down until only one remains.
• Analogy: You have a sculpture with too many arms. You first carve it to a rough shape (Path A), then use a chisel to chop off the extra limbs until only one perfect arm remains (Path B).

The Deep Connection: The "Equivalence Class"

The most surprising discovery in the paper is that all these different paths lead to the same thing.

Whether you use the tipping point, the magnetic peeling, or the hybrid method, the resulting single Weyl fermion is mathematically identical. They all belong to an "Equivalence Class."

• The Metaphor: Imagine you are trying to get to the top of a mountain.
  • Path A is hiking up the north side.
  • Path B is taking a cable car up the south side.
  • Path C is taking a helicopter to a ridge and then hiking.
  • The Discovery: The authors found that no matter which path you take, you end up at the exact same peak, and the view from the top is identical. They are all just different ways of describing the same underlying reality.

They found that all these models can be encoded using a mathematical structure called Spin(4). Think of this as a universal "language" or "DNA" that describes how these particles behave. Whether you are looking at a superconductor or a lattice model, they are just speaking different dialects of the same language.

Why Does This Matter?

1. Solving a 40-Year Mystery: For decades, physicists struggled to put single Weyl fermions on a computer grid (lattice) without creating unwanted duplicates. This paper provides a practical "recipe" to do it.
2. New Materials: It suggests that if we build specific types of superconductors or superfluids, we might naturally find these single particles emerging in the real world.
3. Symmetry is Weird: The paper shows that the "symmetry" (the rules governing the particle) in these systems is non-compact and non-local.
   • Non-local: The rule doesn't just apply to one particle; it connects particles far away from each other instantly.
   • Non-compact: The "charge" isn't a simple integer (like 1 or 2); it can be a continuous value, like a dial you can turn infinitely. This is very different from standard physics rules.

Summary in One Sentence

By breaking the rules of charge conservation in superconductors, the authors found three different ways to "sculpt" a single, isolated Weyl fermion out of a 3D grid, proving that these different methods are actually just different faces of the same mathematical coin.

Technical Summary

1. Problem Statement

The central challenge addressed in this work is the construction of single Weyl fermion lattice models in (3+1) dimensions.

• The No-Go Theorem: The Nielsen-Ninomiya theorem states that in a local, translationally invariant lattice theory with a conserved $U(1)$ charge, Weyl fermions must appear in pairs (doubling), preventing the isolation of a single chiral cone.
• The Conflict: While gapless superconductors and superfluids (which break $U(1)$ symmetry) naturally host nodal points, the connection between these physical systems and the abstract construction of lattice chiral fermions (often used for UV completions of chiral gauge theories) has been unclear.
• Key Questions:
  1. What are the necessary and sufficient conditions to evade the no-go theorem to achieve a single Weyl cone?
  2. Is there a unified framework connecting gapless topological superconductors/superfluids with lattice chiral fermion models?
  3. How are the emergent symmetries at these critical points UV-completed on the lattice?

2. Methodology

The authors employ a Real Fermion (Majorana) formalism to reformulate the problem, moving away from the standard complex fermion description.

• Real Fermion Representation: They demonstrate that any complex fermion system with broken $U(1)$ symmetry can be mapped to a system of real fermions via a unitary transformation (Nambu to Real basis). In this basis, the Hamiltonian satisfies $H(k) = -H^*(-k)$, enforcing intrinsic charge conjugation symmetry.
• Differential Geometry Approach: Instead of traditional Berry curvature analysis, they use intersection theory in differential geometry. They define a 6-dimensional manifold ($T^3 \times \mathfrak{su}(2)$) where band crossings correspond to the intersection of two oriented 3D submanifolds.
  • They prove that for real fermions, intersections always occur in pairs at $\pm k$ due to charge conjugation.
  • They show that a single pair of real fermion crossings ($M=1$) can be projected to form an effective single Weyl cone, whereas even numbers of pairs ($M=2, 4...$) correspond to Dirac fermions.
• Three Construction Paths: The authors explore three distinct routes to generate single Weyl cones from gapped Symmetry Protected Topological (SPT) states (specifically DIII class topological superconductors):
  • Path a): Pushing a gapped SPT with Time-Reversal (T) symmetry to a Topological Quantum Critical Point (tQCP) where the topological invariant changes by its fundamental value ($\delta N_w = 2$).
  • Path b): Applying a T-symmetry breaking field (e.g., magnetic field) to a gapped SPT to "peel off" excessive degrees of freedom, leaving a pair of nodal points that map to a single Weyl cone.
  • Path c): A hybrid approach where tQCPs with larger topological changes ($\delta N_w = 4, 8$) are subjected to T-symmetry breaking fields to reduce the degrees of freedom to a single cone.

3. Key Contributions

A. Evading the No-Go Theorem

The paper establishes that spontaneous breaking of charge $U(1)$ symmetry is a sufficient condition to evade the Nielsen-Ninomiya theorem. By working in the real fermion basis, the authors show that the "doubling" is a consequence of charge conjugation pairing ($\pm k$). A single Weyl cone emerges when these paired real fermion crossings are reinterpreted as particle-hole partners of a single complex fermion.

B. Equivalence Class of Models

The authors prove that all lattice models yielding a single Weyl cone (whether via Path a, b, or c) form an equivalence class.

• Spin(4) Structure: All such Hamiltonians can be encoded in two dual copies of the $(1, 1)$ representation of the Spin(4) group (isomorphic to $SU(2) \times SU(2)$).
• Isomorphism: In the infrared (IR) limit, these models are isomorphic to either:
  1. A T-symmetric tQCP in a DIII class topological superconductor.
  2. Its dual: A T-symmetry breaking superconducting nodal point phase.
• Duality: The two copies of the representation are connected by a $\sigma-\tau$ (spin-charge) duality transformation.

C. UV-Completed Symmetries

A major contribution is the detailed analysis of the UV-completed symmetry groups (conserved charge operators) on the lattice:

• Path a) (T-symmetric tQCP): The conserved charge operators span a 6-dimensional linear space. These charges are non-local (non-on-site) and non-compact.
• Path b) & c) (T-symmetry breaking): The conserved charge operators span a 2-dimensional linear space.
• Non-Compactness: The authors confirm that the symmetry charges in these models are non-compact (continuous spectrum of eigenvalues) and non-local, a feature required to satisfy 't Hooft anomaly matching for emergent single Weyl fermions.

D. Specific Lattice Models

The paper constructs and analyzes five specific lattice models:

• Model I: A generic T-symmetric tQCP ($\delta N_w = 2$).
• Model II: A nodal phase induced by a magnetic field on a gapped SPT.
• Model III: A tQCP ($\delta N_w = 2$) with an added T-breaking field.
• Model IV: A multicritical tQCP ($\delta N_w = 4$) with T-breaking.
• Model V: A multicritical tQCP ($\delta N_w = 8$) with T-breaking, which connects directly to recent proposals for lattice chiral fermions (e.g., Ref [53]).

4. Results

• Phase Boundaries: The authors identify precise phase boundaries in the parameter space (mass $\mu$ and magnetic field $B$) where single Weyl cones emerge, distinguishing them from phases with multiple cones or full gaps.
• Charge Assignment: They demonstrate how to assign conserved charges to the bands such that the two crossing bands at $\pm k$ carry opposite charges, effectively reconstructing a single complex Weyl fermion from two real fermion crossings.
• Dimensional Reduction: They show that the complex 15-dimensional manifold of general lattice Hamiltonians can be projected onto a 9-dimensional sub-manifold (two dual $(1,1)$ representations) specifically for single Weyl fermion models.

5. Significance

• Unification of Fields: The paper bridges the gap between the physics of gapless superfluids/superconductors (where Weyl nodes are common) and lattice chiral fermion constructions (relevant for high-energy physics and quantum simulation). It proves they are different manifestations of the same underlying topological structure.
• Practical Construction: It provides a "recipe" (the three paths) for experimentalists and simulators to engineer single Weyl fermions in cold atom or solid-state systems by manipulating symmetry breaking and topological transitions.
• Symmetry Insights: The discovery of the dimensionality difference in the symmetry charge spaces (6D vs. 2D) depending on T-symmetry provides new constraints for understanding the UV completion of chiral gauge theories.
• Anomaly Matching: The work reinforces the idea that emergent symmetries at tQCPs are anomalous and non-compact, offering a concrete lattice realization of these abstract field-theoretic concepts.

In conclusion, the paper establishes that single Weyl fermions in 3D lattices are not anomalies of specific models but a robust feature of a specific equivalence class of topological states, unified by Spin(4) symmetry and characterized by the spontaneous breaking of $U(1)$ charge symmetry.