Does a level crossing zero prove a quantum 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.

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.

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.

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.

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