How to formulate the $\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice

This paper proposes and numerically verifies a lattice formulation of the $\mathbb{Z}_8$-valued Arf-Brown-Kervaire invariant for Majorana fermions on two-dimensional non-oriented manifolds, demonstrating that it can be extracted from Pfaffians of the Wilson Dirac operator and agrees with continuum theory results.

The Gist

Imagine you are trying to count the number of "twists" in a piece of string, but the string is made of invisible, ghostly particles called Majorana fermions. In the world of quantum physics, these particles are special because they are their own antiparticles.

This paper is about building a digital simulation (a lattice) to count these twists accurately, even when the "string" is shaped like weird, impossible objects like a Möbius strip or a Klein bottle.

Here is the breakdown of their work using simple analogies:

1. The Problem: Counting Twists on a Pixelated World

In the real, smooth world of physics (the "continuum"), we know how to count these twists. It's like counting the knots in a rope. However, when physicists try to simulate this on a computer, they have to turn space into a grid of pixels (a lattice).

• The Issue: When you turn a smooth rope into a pixelated grid, the "knots" can disappear or get distorted. Standard methods for counting these topological features (like the famous "Index Theorem") work well for some particles but fail for Majorana fermions, especially when the shape of the universe is weird (non-orientable).
• The Goal: The authors wanted to create a new rulebook (a formula) that works on this pixelated grid to count a specific type of twist that comes in groups of 8. This is called the $\mathbb{Z}_8$ invariant (or the Arf-Brown-Kervaire invariant).

2. The Solution: The "Ghostly" Mass Switch

To solve this, the authors used a clever trick involving Wilson fermions. Think of this as a special type of particle simulation that doesn't rely on perfect symmetry (which is hard to keep on a grid).

Instead of trying to count the knots directly, they looked at the phase (the "color" or "vibe") of the particle's wave function.

• The Analogy: Imagine you have a giant drum. If you hit it, it makes a sound. The "phase" is like the specific note the drum plays.
• The Trick: They introduced a "mass term" (a heavy weight) to the particles. By turning this weight on and off (or making it very heavy), they could see how the "note" of the drum changed.
• The Result: They found that the note changes in a very specific, quantized way. It doesn't just shift randomly; it jumps between 8 distinct notes (0 to 7). This jump is the invariant they were looking for.

3. The Playground: Weird Shapes

To prove their method works, they didn't just use a flat square (like a normal computer screen). They built simulations of shapes that are impossible in our 3D reality without twisting:

• The Torus (Donut): A standard shape. Easy.
• The Klein Bottle: A bottle with no inside or outside. If you walk along it, you end up on the "other side" of the universe without crossing a boundary.
• The Real Projective Plane (RP2): A shape where if you walk far enough in one direction, you come back to where you started, but you are facing the opposite way (like a mirror image).
• The Möbius Strip: A strip of paper with a half-twist. It has only one side.

The Challenge: On a computer grid, making a Klein bottle or Möbius strip is tricky because you have to "glue" the edges of the grid together in a way that flips the orientation. The authors figured out exactly how to tell the computer to "glue" the pixels with a twist, effectively creating these impossible shapes out of code.

4. The Verification: Does the Math Hold Up?

They ran two types of tests:

1. Analytic (Math): They solved the equations for the flat shapes (Torus and Klein bottle) using pure math.
2. Numerical (Computer): They actually ran the simulation on a computer for various sizes of grids.

The Outcome:

• As they made the grid finer (more pixels, smaller steps), the "notes" the system played settled down perfectly into the expected integers (0, 1, 2... 7).
• Even for the Möbius strip (an open shape with edges), their method worked perfectly, confirming that the "twist" count was correct.

Why Does This Matter?

Think of these topological invariants as the "DNA" of a quantum material.

• If you have a material with a "twist" of 1, it behaves differently than one with a "twist" of 2.
• These materials are called Symmetry Protected Topological (SPT) phases. They are the building blocks for future quantum computers because they are incredibly stable; you can't easily destroy their "twist" without breaking the whole system.

In Summary:
The authors built a new digital ruler that can measure the "twistiness" of ghostly particles on a computer grid. They proved that even when you simulate the universe on a pixelated screen, you can still accurately count these fundamental quantum twists, even on shapes that don't exist in our physical world. This opens the door to simulating and understanding complex quantum materials that could power the next generation of technology.

Technical Summary

Here is a detailed technical summary of the paper "How to formulate the Z8 topological invariant of Majorana fermion on the lattice" by Sho Araki et al.

1. Problem Statement

Topological invariants are essential for understanding non-perturbative phenomena in quantum field theories (QFT), such as anomalies and Symmetry Protected Topological (SPT) phases. While the standard integer-valued Atiyah-Singer index for Dirac operators is well-established on the lattice (via the Ginsparg-Wilson relation and overlap operators), generalizing this to more complex topological invariants remains challenging.

Specifically, certain fermionic systems, particularly Majorana fermions with reflection symmetry on non-orientable manifolds, possess topological invariants valued in $\mathbb{Z}_8$ (the Arf-Brown-Kervaire or ABK invariant). These invariants cannot be expressed simply as the index of a Dirac operator (which counts zero modes) and require a formulation that handles:

• Non-orientable geometries (e.g., Klein bottle, Real Projective Plane).
• The absence of exact chiral symmetry on the lattice.
• The complex phase of the partition function rather than just the magnitude.

The authors aim to formulate a lattice version of the ABK invariant that reproduces the continuum $\mathbb{Z}_8$ classification without relying on the Ginsparg-Wilson relation.

2. Methodology

The authors propose a formulation based on Wilson fermions and the Pfaffian of the massive Wilson-Dirac operator.

A. Theoretical Framework

• Continuum Definition: The ABK invariant ($\beta$) arises in the complex phase of the Majorana fermion partition function $Z$ on a 2D manifold $X$ with a Pin$^-$ structure. In the limit of large negative mass ($m \to -\infty$), the phase is quantized:
  $$Z(X, s; m) \propto \exp\left(i \frac{\pi}{4} \beta\right), \quad \beta \in \{0, 1, \dots, 7\}$$
• Lattice Action: They utilize the massive Wilson-Dirac operator $D_W(m)$ on a 2D square lattice. The partition function is defined as the Pfaffian:
  $$Z = \text{Pf}(C D_W(m))$$
  where $C$ is the charge conjugation matrix.
• Regularization: To extract the topological invariant, they compare the phase of the partition function at a negative mass $m$ against a reference partition function at positive mass $|m|$ (acting as a Pauli-Villars regulator):
  $$\frac{\text{Pf}(C D_W(m))}{\text{Pf}(C D_W(|m|))} \propto \exp\left(i \frac{2\pi}{8} \beta_{\text{latt}}\right)$$

B. Constructing Manifolds on the Lattice

To study non-orientable manifolds, the authors employ twisted boundary conditions (identifying edges with orientation reversal) rather than triangular lattice approaches.

• Torus ($T^2$): Standard periodic/anti-periodic boundary conditions.
• Klein Bottle ($KB$): Gluing edges with orientation reversal in one direction (using reflection operators $R_x, R_y$).
• Real Projective Plane ($RP^2$): Gluing both pairs of edges with orientation reversal.
• Möbius Strip (Open Manifold): Realized using a domain-wall mass term. The mass $m(x,y)$ is negative in a central strip (the Möbius region) and positive outside, effectively creating an open manifold with boundaries where the domain wall fermions reside.

C. Calculation Techniques

1. Analytic Computation: For manifolds with translational invariance (Torus, Klein Bottle), they perform Fourier analysis to compute eigenvalues of $D_W(m)$ and evaluate the Pfaffian product.
2. Numerical Computation: For general cases (including $RP^2$ and the Möbius strip) where momentum decomposition is difficult or impossible, they numerically compute the Pfaffian of the finite-size matrix for varying lattice sizes ($N$) and masses.

3. Key Contributions

• Lattice Formulation of ABK Invariant: The first explicit lattice formulation of the $\mathbb{Z}_8$ ABK invariant using Wilson fermions, bypassing the need for the Ginsparg-Wilson relation or overlap operators.
• Handling Non-Orientable Geometries: A robust method to construct non-orientable manifolds (Klein bottle, $RP^2$) and open manifolds (Möbius strip) on a square lattice using twisted boundary conditions and domain-wall mass terms.
• Pin$^-$ Structure Classification: Explicitly demonstrated how different boundary condition choices (signs of reflection) correspond to the distinct Pin$^-$ structures required for the ABK invariant.
• Numerical Verification: Provided high-precision numerical evidence that the lattice invariant converges to the correct continuum integers as the lattice spacing $a \to 0$ and mass $m \to -\infty$.

4. Results

The authors verified the lattice formulation across four distinct topologies:

Manifold	Pin$^-$ Structure	Continuum $\beta$	Lattice Result ($\beta_{\text{latt}}$)
Torus ($T^2$)	Periodic-Periodic ($PP$)	4	4
	Others ($PA, AP, AA$)	0	0
Klein Bottle ($KB$)	$P\pm$	$\mp 2$	$\approx \mp 2$ (converges rapidly)
	$A\pm$	0	0
Real Projective Plane ($RP^2$)	(Two structures)	$\pm 1$	$\pm 1$
Möbius Strip	(Two structures)	$\pm 1$	$\pm 1$

• Convergence: Numerical data shows that $\beta_{\text{latt}}$ converges to the integer continuum values with high precision even at relatively small lattice sizes (e.g., $N \sim 10$).
• Robustness: The method successfully reproduces results for $RP^2$, a non-flat manifold where the lattice corners introduce curvature singularities, demonstrating the formulation's stability.
• Domain Wall: The domain-wall mass construction correctly captures the topology of the Möbius strip, confirming the connection between open manifolds and edge anomalies.

5. Significance and Outlook

• Non-Perturbative Tool: This work provides a non-perturbative tool to study SPT phases and anomalies in fermionic systems that were previously difficult to simulate on the lattice.
• Extension to Interacting Theories: While focused on free fermions, the authors suggest this framework can be applied to interacting theories. Specifically, they highlight the potential to study the trivialization of SPT phases when the number of Majorana fermions is a multiple of 8 (where the $\mathbb{Z}_8$ anomaly cancels).
• Higher Dimensions: The approach is expected to be extendable to 4D Majorana fermions with Pin$^+$ structures, which possess a $\mathbb{Z}_{16}$ invariant, though this would incur higher computational costs.

In summary, Araki et al. successfully bridge the gap between continuum topological field theory and lattice gauge theory for $\mathbb{Z}_8$ invariants, offering a practical numerical method to explore topological phases in non-orientable settings.