Chiral symmetry is the statement that left-handed and right-handed particles are separately conserved. Quantum mechanics breaks it: no way of regularising the theory preserves both electric charge and chirality at once, and since charge conservation is not negotiable, chirality goes. What is left is an exact statement about how fast it goes.
That equation is famously robust. It comes from a single triangle diagram and yet higher orders cannot change its form — they only replace the bare fields and charge with renormalized ones. In the path-integral language it is not a diagram at all but a property of the measure: the fermion measure is not invariant under a chiral rotation, and the Jacobian is the right-hand side.
There is a second, quite separate way to break chiral symmetry, well known in one dimension: interactions alone will do it, with no electromagnetic field anywhere. This Letter asks what happens when both mechanisms are present at once — which is the situation in an interacting Weyl semimetal.
Regularise with the interaction included
The method is Fujikawa’s, with one change. Fujikawa’s calculation regularises the divergent trace using the eigenfunctions of the free Dirac operator, which diagonalises the action and gives the familiar answer. Here the interaction is first decoupled into an auxiliary field, and the regularisation is done with the Dirac operator that includes it. The auxiliary field is then integrated out at the end rather than dropped at the start.
With and the auxiliary field on shell at , the anomalous term is the usual one built from the total field strength . Integrating out ,
Three terms: one needing only the electromagnetic field, one needing only the interaction, and one needing both. The interactions here are RG irrelevant and would normally be discarded — but in a constant magnetic field they should not be.
Eqs. (6)–(9) · equation numbers throughout are those of arXiv:2102.04371v1
Those three terms can be gathered back into the original shape. Define two dressed fields, each the bare one minus a piece built from the current, and the anomaly equation returns exactly as it was.
Read as screening: for density–density interactions with the electromagnetic field treated semiclassically, and . The anomalous breaking is generated not only by the background fields but by the fluctuations the interacting matter itself induces.
Eqs. (10)–(11)
The same answer from a Luttinger liquid
A magnetic field reduces the dimension of the problem: the anomaly lives on the lowest Landau level, which disperses in one direction only. In the free case that is the standard way to understand the anomaly. Here it works too, provided the lowest Landau level is treated as a Luttinger liquid rather than as free fermions.
Bosonise it, and the modifications appear as they should — with the number of boson flavours identified with the degeneracy of the lowest Landau level. The agreement is not a check of arithmetic so much as an interpretation: the screening factor that appears everywhere in this paper is the charge susceptibility of that Luttinger liquid.
The only assumption is that Landau levels form at all, and that the anomaly lives on a spin-polarized lowest one — reasonable at large field, and at zero field the expression returns the free result. Matching against the bosonized action requires identifying the number of flavours with the Landau-level degeneracy, .
One subtlety the Letter is careful about: the gauge field couples to the fermionic density rather than minimally to the symmetric boson. The excitations of the lowest Landau level are not the bare chiral fermions and carry different electric and chiral charges. Had the gauge field coupled to those instead, the anomaly equation would have come out unmodified.
Eqs. (13)–(15)
What you could measure
In a Weyl semimetal the two nodes are separated in momentum, and rotating that separation away costs a Chern–Simons term — the anomaly again. Varying it gives the anomalous Hall current, and now the dressed field is the one that appears.
The consequence is sharp and it is a null result in the right place: the interaction-dependent piece vanishes in equilibrium, so the equilibrium Hall response is untouched. It is the non-equilibrium and inhomogeneous response that changes. At finite frequency the Hall conductivity acquires a Lorentzian cut-off; and the density response to a change in magnetic field — the chiral separation effect — is screened.
The first term is the quantum anomalous Hall current; the second vanishes in equilibrium. Independently, integrating the fermions out and computing the response within RPA gives the same screening — the longitudinal response of the lowest Landau level is captured exactly by an RPA summation because of its reduced dimensionality, and the transverse components are untouched. The anomalous density response turns out to come not from the polarisation bubble, which vanishes, but from the change in degeneracy of the lowest Landau level.
Eqs. (17)–(21)
This does not contradict Laughlin’s argument that the Hall conductance is insensitive to local interactions: the chiral modes here are not spatially separated as they are on Laughlin’s cylinder. Reproducing this effect there would need non-local interactions between the edges.
The line of work continues in Path-Integral Approach to Quantum Anomalies in Interacting Models, which takes the same method through gravity, spin–spin interactions and a set of massive modes the anomaly makes on its own, and in Non-Abelian Bosonization in a (3+1)-D Kondo Semimetal, where the field the fermions are coupled to is an array of spins rather than an interaction.