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.

Selected positive energy $1.99$

Selected chirality $0.102$

Zone-edge gap $4$

Axial mixing strength $3.96$

A regulated fermion ring

Lift the extra cone

Threaded flux

\Phi/\Phi_0

0.28

Wilson coupling

r

1

Selected momentum mode

n

11

The scan changes the external flux. Dragging the dial pauses it.

Mass and model limits

Bare mass

ma

0

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.

The ring shows a positive-energy Bloch spinor. Spatial phase colors use the uniform-link gauge. Lavender and blue arcs show the two spinor weights, which are uniform across sites.

At an exactly degenerate zero mode, the chirality display uses zero; its eigenbasis is undetermined.

Fujikawa · 2018 · III A

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

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 misconceptionA finite plotting window loses a level, so charge must fail to be conserved.

Better mental modelAll 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

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.
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.
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.

Anomalies & Lattice Spectral Flow

Where the energy levels move