An anomaly is what happens when a symmetry of a classical action cannot be kept by any way of counting the quantum states. In Fujikawa’s way of seeing it, the counting is a choice of basis for the fermion measure, and the anomaly is the Jacobian of a change of variables in that basis. The choice of basis is where an interaction can get in. Regularise with the free Dirac operator and the interaction never appears; regularise with the operator that already contains it, and it appears everywhere.
This paper takes that one change through as many settings as it will go: one spatial dimension, then three; the transport that follows; then the same thing with curvature, a horizon, and interactions that are not between electric currents. What is remarkable is how little the answers vary. Almost everything that comes out is the free result multiplied by one of two numbers.
Built from the interaction strength and the separation of the Weyl nodes. It multiplies every derivative of the current in the anomalous relation.
It is a length.Built from the same interaction and the degeneracy of the lowest Landau level. The anomaly, after a magnetic field has reduced the problem to one dimension, is divided by it.
It is a screening factor.Decouple with an auxiliary field, put the auxiliary field in the Dirac operator, and regularise the Jacobian with that operator rather than the free one. In four dimensions the Ward identity comes out as
Three terms: one needing only the electromagnetic field, one only the interaction, one both. Gathered into dressed fields and it becomes the free relation again — which is the result of the earlier Letter, re-derived here as one case among several.
Eqs. (4.10)–(4.12) · equation numbers throughout are those of arXiv:2302.14191v1
One dimension comes first, because there the whole calculation can be done by hand and checked. An interaction between currents renormalises the anomaly by a factor , and the excitations acquire a modified mass. That the four-dimensional calculation reproduces this after a magnetic field has reduced it to one dimension is the paper’s internal check.
The first number is a length
In an interacting Weyl semimetal the anomalous current obeys a relation that involves not only the electromagnetic field but the current itself, one derivative down and one factor of in front. That single fact has two consequences, and the second one is not obvious at all.
The first: the equilibrium anomalous Hall response is untouched, because the extra term needs a time derivative or a gradient to be non-zero. Away from equilibrium it bites. The Hall conductivity acquires a Lorentzian cut-off at — and a longitudinal conductivity appears that has no business being there at all, made entirely out of the interplay between the interaction and the Hall response.
The second consequence is the one worth the page. Switch the electromagnetic field off altogether and the anomalous current does not go quiet. It still satisfies a relation that ties it to its own derivatives, and that relation, used on itself, is a wave equation.
With , and , the anomalous current obeys . The component along the node separation vanishes; two of the remaining three equations carry time derivatives and the third is a constraint. Feeding the first two into the third, and using the conservation the Levi-Civita symbol guarantees, leaves
Three Klein–Gordon equations, one for each surviving component. So : is the reduced Compton wavelength of these waves, and restoring and gives their mass, . Restore the electromagnetic field and it appears on the right-hand side as a source: . The role of the speed of light is played by , which is why the effect is not the same as shining light on a non-interacting sample.
Eqs. (7.1)–(7.9)
These are excitations that exist because the anomaly does. They are not perturbative — the mass goes as one over the interaction strength, so it is large exactly where perturbation theory would be safe, and it goes to zero when the interaction is strong or the Weyl nodes are far apart. In that limit the waves run at the Fermi velocity.
The second number is a screening
A magnetic field reduces the problem: the anomaly lives on the lowest Landau level, which disperses along one direction only, and the one-dimensional calculation applies. What comes back is the anomalous relation divided by , with the degeneracy of that level. The same factor screens the density response to a change in magnetic field, and it can be read as the charge susceptibility of the Luttinger liquid the lowest Landau level has become.
So far this is the earlier Letter, recovered. The new part is where the factor goes next.
The same factor, in a gravitational field
Curvature adds a term to the anomaly of its own: alongside sits the Pontryagin density of the Riemann tensor, . It is there for the same reason the electromagnetic term is — the Dirac operator in curved space carries a spin connection, and it is that operator which regularises the measure.
Current–current interactions produce no cross terms with the curvature. After the same dimensional reduction, and for , both terms end up divided by the same factor:
Eqs. (8.1) and (8.4)
needs a twist in the geometry, not merely curvature: a spherically symmetric geometry gives zero. The paper works two line elements out explicitly — a region of twisted angle, and a spiral along an axis — to show what the density is measuring.
Now take a geometry with a horizon: twisted, with a non-zero Pontryagin density, and flat far away. Near the horizon the physics reduces to a chiral theory in one dimension. Integrating the anomalous relation outward, with the boundary conditions that the flux is finite at the horizon and the geometry is flat at infinity, leaves a chiral current escaping to infinity that is fixed entirely by the surface gravity.
which is the temperature part of the chiral vortical effect, arrived at from the gravitational anomaly. With the interaction switched on from the start, the factor rides along and never leaves:
Eqs. (8.7)–(8.8) and the two that follow them
There are two ways to read that last line and the paper offers both. Either the interactions change how much charge flows along the axis, or the temperature of the radiation is not simply the surface gravity over once the matter interacts. The second reading is the more provocative one, and it is the same factor that started as the charge susceptibility of a Landau level.
This is testable in the direction it came from. Mixed axial–gravitational anomalies have been invoked to explain thermal transport measurements in Weyl semimetals. If those measurements really are the gravitational anomaly, they should carry this modification. If the modification is looked for and is not there, the identification is the thing in question.
Interactions that are not between currents
Everything above is for a local interaction between electric currents. Not every interaction is of that kind: an interaction between chiral currents — whose spatial part is the spin–spin interaction between Dirac fermions — is equivalent to it in two dimensions but not in four. Run the same procedure on it and the auxiliary field enters the Dirac operator with a , so the transformation that decouples it is a different one and the anomalous term it leaves is different too. The method does not care; the answer does.
The three papers of this line read in order: Chiral Anomaly in Interacting Condensed Matter Systems finds the interacting anomaly and its measurable consequences; Non-Abelian Bosonization in a (3+1)-D Kondo Semimetal turns the anomaly into a calculational tool for a coupled spin system; and this one takes the method as far as it goes.