---
title: "Anomalies & Lattice Spectral Flow Interactive Lab | Sandbox Physics"
description: "What happens to the extra fermion at the edge of a lattice spectrum? The Wilson term moves the extra lattice fermion to high energy by mixing its chiral components. The low-momentum species remains light at zero bare mass. This exposes how the regulator treats high energies; a continuum axial-anomaly calculation also needs a specified regulated limit."
canonical_url: "https://sandboxphysics.com/labs/anomaly-spectral-flow/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/anomaly-spectral-flow/"
last_modified: "2026-10-27"
---

FR02 · Follow the high-energy states

# Anomalies & Lattice Spectral Flow

**What happens to the extra fermion at the edge of a lattice spectrum?** The Wilson term moves the extra lattice fermion to high energy by mixing its chiral components. The low-momentum species remains light at zero bare mass. This exposes how the regulator treats high energies; a continuum axial-anomaly calculation also needs a specified regulated limit.

Thread a ring with flux and follow its energy levels. A Wilson term lifts the extra low-energy cone; watch how the same term mixes the two chiral components.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

A free two-component fermion on a finite antiperiodic ring, with continuous time and a uniform external U(1) link. The display samples instantaneous eigenstates; it does not evolve occupations. The Wilson term explicitly breaks axial symmetry at finite spacing. The continuum axial anomaly needs a regulated limit.

## Where the energy levels move

Default spectrum saved

Momentum $pa$ · Energy $Ea$

Curves: infinite-lattice reference. Dots: current finite-ring spectrum. Gold rings: selected mode. The export keeps the last saved spectrum.

[Read the course: gauge fields and effective theories](https://sandboxphysics.com/stories/gauge-effective-theory/)

Physics tutorial

## Anomalies & Lattice Spectral Flow

**Background** Thread a ring with flux and follow its energy levels. A Wilson term lifts the extra low-energy cone; watch how the same term mixes the two chiral components.

**Why it matters** Replacing a derivative by a nearest-neighbour difference creates an extra low-energy fermion. Its fate matters when a lattice calculation tries to describe one continuum species.

Start with the essentials

- **Focus question**: What happens to the extra fermion at the edge of a lattice spectrum?
- **One-sentence intuition**: The Wilson term lifts the extra cone by mixing chiral components at high momentum.

### Core mathematical model

#### A two-component lattice Hamiltonian

$$
\begin{aligned}aH(p)&=\sigma_3\sin(pa)+\sigma_1 B(p),\\B(p)&=ma+r(1-\cos(pa))\end{aligned}
$$

The lattice spacing sets the length unit; Planck’s constant and velocity are one. The Pauli matrices act on the two components. The bare mass is nonnegative here. At zero Wilson coupling and zero mass, the sine vanishes both at the origin and the zone edge.

#### Flux shifts the allowed covariant momenta

$$
\begin{aligned}p_na&=\frac{2\pi}{N}\left(n+\frac12-\frac{\Phi}{\Phi_0}\right),\\ E_\pm a&=\pm\sqrt{\sin^2(p_na)+B(p_n)^2}\end{aligned}
$$

The half step imposes an antiperiodic boundary condition. One flux quantum permutes the complete set of levels. The display wraps momenta into the first Brillouin zone.

#### Why axial symmetry changes

$$
\begin{aligned}\gamma_5&=\sigma_3,\\\left\|[aH,\gamma_5]\right\|&=2\left|B(p)\right|\end{aligned}
$$

The operator norm measures explicit mixing at this momentum. With Wilson coupling one and zero mass it is small near the origin and large at the zone edge. This finite-spacing commutator is distinct from the continuum anomaly.

### Common difficulties

#### A missing level is not an anomaly proof

**Typical misconception** A finite plotting window loses a level, so charge must fail to be conserved.

**Better mental model** All ring momenta remain in the calculation. The vector U(1) charge is conserved. To discuss the axial anomaly, specify how the occupied sea is regulated and how the continuum limit is taken.

### Run the experiment

1. 01

   #### Find both zeroes

   Choose “The extra fermion” and pause the scan. Compare the center and both zone edges in the spectrum.

   **What to observe:** Both regions have low-energy levels. The selected high-momentum mode is the extra species.

2. 02

   #### Lift the extra cone

   Increase Wilson coupling continuously. Follow the zone-edge gap and the chirality bar.

   **What to observe:** The high-momentum levels move apart and their components mix. The low-momentum cone stays light when the bare mass is zero.

3. 03

   #### Compare a full flux cycle

   Save spectra at zero and one flux quantum and export each. Sort the energies before comparing them.

   **What to observe:** The sorted spectra agree. Mode labels move by one, as required by large gauge invariance.

Physical conclusion

A free two-component fermion on a finite antiperiodic ring, with continuous time and a uniform external U(1) link. The display samples instantaneous eigenstates; it does not evolve occupations. The Wilson term explicitly breaks axial symmetry at finite spacing. The continuum axial anomaly needs a regulated limit.

## Related experiments and articles

Gauge fields and effective theories · Reading path

### Gauge symmetry, effective theories and anomalies

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Does a level crossing zero prove a quantum anomaly?**](https://sandboxphysics.com/stories/gauge-effective-theory/anomaly-spectral-flow/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)