Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions

This paper constructs 3+1D and 2+1D Hamiltonian lattice models for Weyl fermions and Dirac cones that evade no-go theorems by utilizing exact, not-on-site chiral symmetries—specifically a Ginsparg-Wilson analog and an Onsager algebra—to protect gaplessness and generate global anomalies even in the absence of crystalline translations.

The Gist

The Big Problem: The "Fermion Doubling" Curse

Imagine you are an architect trying to build a digital city (a computer simulation) that perfectly mimics a specific type of particle called a Weyl fermion. These particles are special because they are "chiral," meaning they only spin in one direction, like a screw that can only be turned clockwise.

For decades, physicists have been stuck on a major construction problem known as the Nielsen-Ninomiya Theorem (or the "Fermion Doubling" problem).

The Analogy:
Think of the digital city as a grid of tiles (a lattice). You want to place a single, perfect "clockwise screw" on this grid. However, the laws of physics governing grids are tricky. Every time you try to place one clockwise screw, the grid forces you to accidentally create a "counter-clockwise screw" right next to it.

In the real world, you might want just one type of particle. But on a computer grid, they always come in pairs (one left-handed, one right-handed). It's like trying to paint a wall red, but the paint bucket is cursed: every drop of red paint you pour instantly creates a drop of blue paint next to it. You can never get a purely red wall.

The Old Solutions: "Gapping" and Breaking Rules

Previously, physicists tried to fix this by:

1. Adding Mass: Trying to "glue" the unwanted counter-clockwise screw to the floor so it stops moving (giving it mass). But to do this, they had to break the rules of the city (symmetries), which made the simulation unstable and required fine-tuning.
2. Wilson Fermions: A method that adds a heavy "weight" to the unwanted particles to stop them, but this breaks the very symmetry you were trying to protect.

The New Solution: "Not-On-Site" Symmetries

The authors of this paper, Lei Gioia and Ryan Thorngren, found a clever loophole. They realized that to get a single, perfect Weyl fermion, you don't need to break the rules; you just need to change how the rules are applied.

The Analogy:
Imagine a security guard (Symmetry) checking IDs at a club.

• Old Way (On-Site): The guard checks the ID of the person standing right in front of him. If the person is a "clockwise screw," the guard lets them in. If they are a "counter-clockwise screw," the guard stops them. But the grid forces the counter-clockwise screw to be there, so the guard has to break the rules to stop it.
• New Way (Not-On-Site): The guard doesn't just look at the person in front of him. He looks at the person in front of him AND the person standing one step to the left. He checks a relationship between neighbors.

By looking at the neighborhood (not just the single spot), the guard can create a rule that allows the "clockwise screw" to exist perfectly while naturally canceling out or "gapping" the unwanted "counter-clockwise screw" without breaking any laws.

The Two Main Models They Built

The paper presents two specific blueprints for these new cities:

1. The Single Weyl Fermion (The "Magic Wire")

• The Goal: Create a model with exactly one Weyl fermion.
• The Trick: They used a symmetry that is "non-compact."
• The Metaphor: Imagine a standard lock that only has a few specific keys (quantized). This new lock has a dial that can spin infinitely (non-quantized). Because the dial can spin anywhere, it can perfectly balance the system to keep only one type of particle.
• Why it matters: This is the first time someone has built a "ultralocal" model (where interactions only happen between immediate neighbors, keeping the simulation fast and simple) that achieves this. Previous models required looking at the whole city at once, which is computationally impossible.

2. The Weyl Doublet (The "Onsager Dance")

• The Goal: Create a model with a pair of Weyl fermions that act like a team (a doublet).
• The Trick: They combined two types of symmetries. One is a standard "on-site" check, and the other is a "neighbor" check.
• The Metaphor: Imagine two dancers. Individually, they have their own moves. But when they dance together, they don't just do a simple waltz (which would be a standard symmetry). Instead, they perform a complex, infinite dance known as the Onsager Algebra.
• The Result: This complex dance protects the pair of particles. Even if you try to break the symmetry of the city (like breaking the grid's translation rules), the dancers are so tightly linked by this complex dance that they cannot be stopped or given mass. They remain "gapless" (massless).

Why This is a Big Deal

1. Beating the "No-Go" Theorems: There were mathematical proofs saying "You cannot do this." The authors showed those proofs assumed you were only looking at single spots (on-site). By looking at neighbors (not-on-site), they bypassed the proof.
2. Simplicity: These models are "ultralocal." This means they are simple enough to actually run on a computer to simulate complex physics, like the Standard Model of particle physics.
3. Condensed Matter Connection: These models aren't just math; they describe real materials called Weyl Semimetals. The authors show that these materials have a hidden "exact symmetry" that protects their special electronic properties, even if the material is imperfect.

The Takeaway

The authors have built a new kind of "digital playground" for particles. By changing the rules from "checking the person alone" to "checking the person and their neighbor," they managed to isolate a single, perfect particle that was previously thought impossible to isolate on a grid.

It's like finally finding a way to paint a wall purely red without the cursed blue paint appearing, by realizing you need to paint the wall and the floor together as a single unit. This opens the door to simulating the fundamental forces of the universe with much higher precision than ever before.

Technical Summary

1. Problem Statement

The central problem addressed is the fermion doubling problem in lattice field theory, specifically the difficulty of regulating chiral gauge theories (like the Standard Model) on a lattice without introducing unphysical fermion species.

• Nielsen-Ninomiya Theorem: This theorem states that for any lattice system of free fermions with an on-site $U(1)$ symmetry, the low-energy (IR) theory must contain equal numbers of left- and right-handed Weyl fermions. This "doubling" prevents the isolation of a single chiral fermion.
• Current Limitations: Existing solutions often involve:
  • Wilson fermions: Introduce mass terms that break chiral symmetry, requiring fine-tuning to restore it in the continuum limit.
  • Ginsparg-Wilson (GW) symmetry: A non-on-site symmetry in Euclidean lattice models that protects chiral symmetry. However, standard GW constructions (e.g., overlap fermions) are not ultralocal (finite-range in real space), making them computationally expensive and difficult to implement in Hamiltonian formulations.
• Goal: The authors aim to construct ultralocal (finite-range hopping) Hamiltonian models in 3+1D and 2+1D that host minimal numbers of Weyl/Dirac fermions protected by exact, non-on-site chiral symmetries, thereby evading no-go theorems without fine-tuning.

2. Methodology

The authors utilize the Bogoliubov-de Gennes (BdG) formalism to construct tight-binding models.

• BdG Formalism: They treat the system using a basis that includes both particle ($c_k$) and hole ($c_{-k}^\dagger$) operators. This allows for the construction of Hamiltonians that satisfy particle-hole redundancy while introducing symmetry-breaking terms to gap out unwanted nodes.
• Not-on-Site Symmetries: Instead of on-site symmetries (which act only on $\psi(x)$), they construct not-on-site symmetries. These generators involve couplings to neighboring sites (finite range), making them "ultralocal" but non-local in the strict on-site sense.
• Anomaly Matching: They leverage 't Hooft anomaly matching arguments. If a symmetry is anomalous in the IR, the UV lattice theory must possess a corresponding structure to support it.
• No-Go Theorems: They prove new no-go theorems (extending Fidkowski and Xu) showing that a chiral $U(1)$ symmetry with an exponentially local and quantized charge operator cannot protect a single Weyl fermion. This necessitates either non-quantized charges or non-ultralocal operators.

3. Key Contributions and Results

A. Single Weyl Fermion in 3+1D (Time-Reversal Broken)

• Model: A 3+1D magnetic Weyl semimetal model on a cubic lattice with two bands per site.
• Construction:
  • Starts with a model hosting two Weyl nodes at $k=0$ and $k=(0,0,\pi)$.
  • A $U(1)$ symmetry-breaking term is added to gap the node at $(0,0,\pi)$.
  • The remaining single Weyl node at $k=0$ is protected by a new symmetry generator $S_{\text{chiral}}(k)$.
• Symmetry Properties:
  • The charge operator $\hat{Q}_{\text{chiral}}$ is not-on-site (involves nearest-neighbor couplings).
  • Crucially, the spectrum of the generator is non-quantized (continuous, $\mathbb{R}$ action), unlike standard $U(1)$ charges.
  • The generator acts as a bundle of decoupled Majorana chains along the $z$-axis.
• Significance: This evades the Fidkowski-Xu no-go theorem because the charge is not quantized. It provides the first ultralocal Hamiltonian model for a single Weyl fermion with an exact chiral symmetry.

B. Double Weyl Fermion (Weyl Doublet) in 3+1D

• Model: A model hosting a pair of Weyl fermions (a doublet) with opposite chirality.
• Symmetry Structure:
  • The model possesses two exact $U(1)$ symmetries: one on-site ($\hat{Q}_0$) and one not-on-site ($\hat{Q}_1$).
  • Individually, these are anomaly-free. However, together they generate an Onsager algebra (an infinite-dimensional Lie algebra) rather than a standard $SU(2)$ algebra in the UV.
  • In the IR (low energy), this algebra reduces to the global $SU(2)$ flavor symmetry of the doublet.
• Anomaly Protection: The system exhibits Witten's global $SU(2)$ anomaly. This anomaly protects the gaplessness of the Weyl nodes even if crystalline translation symmetry is broken, provided the Onsager algebra structure is maintained.
• Significance: This demonstrates how exact lattice symmetries can generate emergent continuous symmetries with non-trivial anomalies in the IR, realizing the celebrated magnetic Weyl semimetal in a rigorous Hamiltonian framework.

C. Single Dirac Fermion in 2+1D

• Model: A 2+1D model with a single Dirac cone.
• Symmetry: Protected by a $U(1) \rtimes \mathcal{T}$ (parity) anomaly, where $\mathcal{T}$ is time-reversal symmetry.
• Result: The authors construct an exact symmetry-protected single Dirac cone. The symmetry generator is non-quantized and not-on-site, similar to the 3+1D single Weyl case.

D. No-Go Theorem for Quantized Local Symmetries

• The authors prove that for free fermions, a chiral $U(1)$ symmetry with an exponentially local and quantized charge operator cannot act non-trivially on a single Weyl fermion in the IR.
• Implication: To have a single Weyl fermion, one must either:
  1. Use a non-quantized charge (as in their 3+1D single Weyl model).
  2. Use non-ultralocal (long-range) operators (as in Appendix C/D for time-reversal invariant models).

4. Significance and Implications

• Resolution of the Doubling Problem: The paper provides a rigorous "sharp" solution to the fermion doubling problem for free fermions by utilizing not-on-site symmetries. It shows that the Nielsen-Ninomiya theorem's constraints on on-site symmetries can be bypassed without fine-tuning.
• Ultralocality: Unlike overlap fermions (which are non-ultralocal), these models are ultralocal (finite range in real space), making them potentially more amenable to numerical simulation and physical realization in condensed matter systems (e.g., engineered Weyl semimetals).
• Emergent Symmetries: The work clarifies the relationship between UV lattice symmetries and IR emergent symmetries. It introduces the concept of "emanant symmetries" (after Seiberg), where the UV symmetry (Onsager algebra) forces the existence of a specific IR anomaly ($SU(2)$), protecting the gapless phase.
• Condensed Matter Relevance: The models map directly to Weyl semimetals. The results suggest that exact chiral symmetries in these materials can be understood as specific not-on-site lattice symmetries, offering new perspectives on topological protection mechanisms.
• Future Directions: While these models protect chiral fermions, the authors note that gauging these not-on-site symmetries remains difficult due to the non-vanishing 't Hooft anomalies. This presents a challenge for constructing lattice models of chiral gauge theories (like the Standard Model) but opens a new avenue for understanding the interplay between locality, symmetry, and anomalies.

Summary of Technical Innovations

Feature	Traditional Approaches	This Paper's Approach
Symmetry Type	On-site $U(1)$ (leads to doubling) or Ginsparg-Wilson (Euclidean, non-ultralocal)	Not-on-site (finite range) but Ultralocal
Charge Quantization	Quantized (Integer)	Non-quantized (Continuous spectrum) for single Weyl
Algebra	Standard Lie Algebras	Onsager Algebra (Infinite-dimensional) for doublet
Protection	Fine-tuning or emergent symmetry	Exact symmetry protection against mass terms
Locality	Non-local (Overlap) or On-site	Ultralocal (Finite range hopping)

This work fundamentally shifts the paradigm for lattice fermions by demonstrating that exact chiral protection is possible in ultralocal Hamiltonians, provided one relaxes the requirement of on-site quantization.