---
title: "Does a level crossing zero prove a quantum anomaly?"
description: "Recalculate a one-dimensional Wilson-fermion spectrum and distinguish doubling, explicit chiral breaking, spectral flow and a continuum anomaly."
canonical_url: "https://sandboxphysics.com/stories/gauge-effective-theory/anomaly-spectral-flow/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/gauge-effective-theory/anomaly-spectral-flow/"
last_modified: "2026-10-26"
---

[Gauge theory and EFT](https://sandboxphysics.com/stories/gauge-effective-theory/)

L4 · Chapter 7

# Does a level crossing zero prove a quantum anomaly?

Sandbox Physics · 2026-10-26

Flux-dependent levels show how low-energy modes connect to a regulator. Interpreting them as a quantum anomaly also requires vacuum occupations, conserved currents, regularization and a continuum limit. A finite spectral crossing alone specifies a property of the spectrum.

## Before you begin

Separate advanced branch: complex linear algebra, Pauli matrices, single-particle spectra, fermion filling, antiparticles and periodic boundaries. Chapters 1–2 establish the gauge conventions. The mediator fits of chapters 4–6 are not prerequisites.

[Review the preceding prerequisite](https://sandboxphysics.com/stories/gauge-effective-theory/covariant-derivative-ward/)

- Calculate lattice energies and chirality expectations from a two-by-two Hamiltonian.
- Locate the second low-energy point of naive discretization.
- Identify the missing steps between a browser spectrum scan and a continuum anomaly.

## Begin with a linear spectrum

A one-dimensional massless Dirac particle has two linear branches. For $H(p)=\sigma_3p$, one component’s energy rises with momentum and the other falls. In this convention $\sigma_3$ labels the two chiral components and their propagation directions.

A field-theory vacuum also needs a filling prescription: which negative-energy states are occupied, and how particles and holes are defined relative to them. Two plotted lines do not provide that information. Spectral flow tracks levels as a parameter changes; particle production requires the actual state evolution.

Source: David Tong

## A lattice derivative introduces a second low-energy point

On a chain of spacing $a$, the symmetric derivative becomes $\sin(ap)/a$. Near zero momentum this approaches $p$, but it vanishes again at the Brillouin-zone edge $p=\pi/a$. This additional low-energy mode is fermion doubling.

Increasing the number of plotted momentum samples does not remove it. The second zero belongs to the periodic derivative. A different regulator changes how the edge modes enter the low-energy description.

$$
H_{\rm naive}(p)=\sigma_3\frac{\sin(ap)}{a},\qquad E_{\pm}(p)=\pm\frac{\sin(ap)}{a}
$$

Here the signs label fixed chirality components. Sorting by upper and lower energy instead gives plus and minus the absolute value. Spectral-flow labels must specify the choice.

## A Wilson term lifts the edge mode

Read Kazuo Fujikawa, Physical Review D 97, 016018 (2018), section III A, equations (45) and (50), PDF page 10. The Wilson term is second order at low momentum and large at the zone edge. Our teaching extension adds a tunable mass $m$ and sets $a=\hbar=v=q=1$. Energy is measured in $\hbar v/a$ and momentum in $1/a$.

Set $b(p)=m+r(1-\cos p)$. Anticommuting Pauli matrices make the Hamiltonian squared a scalar times the identity, giving its eigenvalues directly. The upper state has chirality expectation $\sin p/|E|$; the lower state has the opposite value. At an exact zero-energy degeneracy, the eigenbasis retains a choice of chirality.

$$
H(p)=\sigma_3\sin p+\sigma_1\bigl[m+r(1-\cos p)\bigr]
$$

Equation (45) is the zero-mass case. The mass control is an explicit teaching extension.

$$
H^2=\bigl[\sin^2p+b(p)^2\bigr]I,\qquad E_\pm=\pm\sqrt{\sin^2p+b(p)^2}
$$

The anticommutation identity provides a check independent of numerical diagonalization.

Analytic limits of the continuous momentum symbol at zero mass. These are model references, not finite-ring observations.

| Regulator | $p=0$ | $p=\pi/2$ | $p=\pi$ |
| --- | --- | --- | --- |
| Naive $r=0$ | $E=0$ | $E=\pm1$ | $E=0$ |
| Wilson $r=1$ | $E=0$ | $E=\pm\sqrt2$ | $E=\pm2$ |

Source: Kazuo Fujikawa

## Insert flux through a finite ring

Use a ring of $N$ sites with antiperiodic fermion boundary conditions, producing a half-integer momentum shift. A uniform link phase represents total flux $\varphi$ in radians. Define $\varphi=qLA$ with $L=Na$. Minimal coupling gives the sampled momenta below.

Increasing $\varphi$ by $2\pi$ shifts the momentum labels by one place and restores the complete spectral set. A labeled point can move into its neighbor’s place even though the final set is unchanged.

The browser calculates instantaneous energies and chirality expectations. Autoplay scans a background parameter. It does not solve the time-dependent Schrödinger equation or determine occupations and transitions. Saved samples contain instantaneous spectra and chirality; particle numbers additionally require occupation dynamics.

$$
p_n(\varphi)=\frac{2\pi(n+1/2)-\varphi}{N},\qquad n=0,\ldots,N-1
$$

Momentum is identified periodically in the Brillouin zone. Antiperiodic boundaries avoid a forced zero-momentum sample at zero flux; a scan can still pass through degeneracies.

## What the continuum axial anomaly additionally needs

Vector charge counts particles minus antiparticles. Axial charge also distinguishes the two chiral directions. A continuous massless theory that preserves vector gauge invariance can have an anomalous axial current. A gauge anomaly would instead obstruct the consistency required of the gauge theory.

For a continuous linear spectrum, the state density per unit length and momentum is $1/(2\pi)$. An electric field drives $\dot p=q\mathcal E$. The two opposite crossing contributions give axial-density rate $q\mathcal E/\pi$. With $\mathcal E=-\partial_t A$ and $\varphi=qLA$, the result is the flux relation below. It uses a filled continuum Dirac sea and vector-current-preserving regularization as a separate reference argument.

$$
\frac{dQ_5}{dt}=\frac{qL\mathcal E}{\pi}=-\frac{\dot\varphi}{\pi},\qquad \Delta\varphi=2\pi\ \Longrightarrow\ \Delta Q_5=-2
$$

Axial charge is right-moving minus left-moving, and positive flux lowers kinetic momentum. Charge, axial and flux conventions determine the sign together. This is not a finite-ring scan readout.

Source: David Tong · Kazuo Fujikawa

## Test an anomaly interpretation

The Wilson term explicitly breaks the simple chiral symmetry already at finite spacing: $[H,\sigma_3]=-2i b(p)\sigma_2$. A mass also contributes to this commutator. A changing simple chiral quantity must first be checked against these explicit Hamiltonian terms.

Fujikawa’s section III B distinguishes finite-depth spectral flow from regulated anomalies. A continuum argument must track high-momentum contributions, the filled vacuum, current definitions and the order of limits. Restricting attention to half the zone and interpreting discarded modes as lost charge confuses a window with a physical result.

Begin with reproducible spectral checks: permutation after a flux cycle, the low-momentum dispersion and the edge gap. A subsequent particle-production calculation needs an initial state and time-dependent evolution, together with conservation of the complete vector charge.

Source: Kazuo Fujikawa

## Try the reasoning in an experiment

### [Keep the low-energy mode and regulator together](https://sandboxphysics.com/labs/anomaly-spectral-flow/)

1. Predict which extra low-energy point appears when the Wilson term is removed at zero mass.
2. Compare naive and Wilson presets. Record low-momentum and edge gaps, then scan one flux cycle and check the complete spectral set.
3. Save instantaneous spectral samples and identify their energies and chirality expectations. Explain which initial-state and evolution inputs are missing from any claim about produced particle numbers.

## Exercises and explanations

1. Why does a massless Wilson term preserve the leading low-momentum linear dispersion?

   Read the explanation

   Because $\sin p=p+O(p^3)$ and $1-\cos p=p^2/2+O(p^4)$. The Wilson contribution is smaller near zero but reaches $2r$ at the edge.

2. With $r=1$ and mass $0.2$, what are the positive energies of the continuous symbol at zero and edge momentum?

   Read the explanation

   They are $0.2$ and $2.2$. The low-energy point is gapped and the edge retains a larger regulator gap. Whether a finite ring samples these exact momenta depends on flux and boundary conditions.

3. Why can a restored spectral set still have nontrivial flow?

   Read the explanation

   The spectrum is an unordered set; flow follows labeled levels and their permutation. Occupations must additionally be tracked to infer particle numbers.

[Previous: Does more precise data make a low-energy theory more reliable?](https://sandboxphysics.com/stories/gauge-effective-theory/truncation-and-positivity/)

## Sources and reproduced scope

Derivations, tables and the plot are authored for this course. The reading locations below separate source results from our teaching extensions. Numerical examples are model references or labeled synthetic observations. No experimental dataset or source figure is copied.

1. [David Tong, Gauge Theory (2018 lecture notes)](https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf)

   Sections 2.1–2.1.3 (connections and Wilson loops), 2.7.3 (Higgs phase), 3.1.1–3.2.2 (two-dimensional anomaly and Ward identities), 4.2.1 (link endpoints) and 4.3.1–4.3.3 (lattice fermions). Printed pages 26–37, 105–108, 122–139, 207–210, 220–228.

2. [Kazuo Fujikawa, Characteristics of chiral anomaly in view of various applications, Physical Review D 97, 016018 (2018), DOI 10.1103/PhysRevD.97.016018](https://journals.aps.org/prd/pdf/10.1103/PhysRevD.97.016018)

   Section III A, equations (45)–(50), PDF page 10, defines the Wilson regulator and doubling spectrum; section III B discusses finite-spacing spectral flow. The tunable mass and finite antiperiodic ring are our stated extensions. The publisher licenses the paper under CC BY 4.0; no original figure is copied.