Non-Abelian Bosonization in a (3+1)-D Kondo Semimetal via Quantum Anomalies

In one dimension there is a way out of a coupled problem. Rotate the fermions by a well-chosen position-dependent phase and the field they are coupled to falls out of their equation; what is left is free fermions on one side and that field, alone, on the other. The rotation is not free. Inside a path integral it has a Jacobian, and the Jacobian is where the interacting physics went. Calculating it is a job for the chiral anomaly.

That is functional bosonization, and in one dimension it can be carried through to the end. This paper asks for it in three dimensions and one of time, for Dirac fermions coupled to an array of localised spins — a Kondo semimetal, where the Kondo effect leaves Weyl fermions as the low-energy excitations. In high dimensions there is no Bethe ansatz and no conformal field theory to fall back on, and the usual move is a slave particle. This is a different move.

For physicists

Four-component Dirac fermions, an electromagnetic field , and an arbitrary spin field at exchange strength . Nothing is assumed about the spins’ own action, and may be classical or quantum. The same action describes a semimetal under strain, where the strain-induced chiral gauge field takes the place of the spins — a rotational strain gives a chiral magnetic field .

Eq. (2) · equation numbers throughout are those of arXiv:2107.12388v2

Three moves

When is constant, one chiral rotation does it: send and the spins vanish from the action. This is the standard manoeuvre for a Weyl semimetal, and its Jacobian is an axion-like term whose variation gives a Hall current along and a density along .

When varies from place to place a phase is no longer enough, because the rotation now has to act on the spinor index as well. The transformation that works has three exponents rather than one, and the paper gives a way to see why.

before

An array of spins of arbitrary length and orientation, one at every point.

move one

A Lorentz rotation, done locally, turns them all the same way. Their lengths still differ. Costs nothing.

move two

A Weyl rescaling makes them all the same length. Now they are one constant field. Anomalous.

move three

A chiral rotation removes a constant field, and the fermions are free. Anomalous.

Two of the three moves leave a Jacobian behind. The Lorentz rotation does not; the Weyl rescaling and the chiral rotation each do, and the sum of the two is everything this paper computes.

For physicists

with . The three real functions , and parameterise the chiral, Weyl and Lorentz parts, and they are fixed by demanding that the spin term disappear:

After the rotation, a change of coordinates whose Jacobian cancels the Weyl factor leaves free Dirac fermions coupled to a gauge field seen in the rotated, rescaled frame. Notice what the defining equation is: the same Dirac equation the untransformed fermion obeys. is the operator taking the field from the Heisenberg to the interaction picture — carried out here inside the path integral, where it turns out to be anomalous.

Eqs. (4)–(10)

The manoeuvre, twice. A spin field over a plane with electrons moving in it; then the interaction term; then the same three moves seen two ways. In the passive picture the local frames turn and the spins are left alone. In the active picture the frames are held and the spins are turned parallel, rescaled equal, and removed. Both are the same transformation; only the bookkeeping differs. Animated for the thesis defence.

Finding the frame

The defining equation is nonlinear, so it is solved by expanding in the exchange strength. What comes out at leading order is a surprise worth stating plainly: the equations for the decoupling parameters are Maxwell’s equations, with the spin texture as the source.

For physicists

With and , and everything expanded as , the first order reads

These are Maxwell’s equations with magnetic source terms, and their solution is the one classical electromagnetism already knows: , , , with the Green’s function of the d’Alembertian. And : at this order no Weyl rescaling is needed at all, and the magnetic source terms drop out with it. Written through the Dirac Green’s function , the whole first-order transformation is one line,

Eqs. (12)–(15)

Each order carries at least one more derivative of than the last. So this is a gradient expansion: it can be truncated for long-wavelength physics, and for a constant texture it terminates at the first term, exactly.

The bill

Now the cost. Run the transformation not all at once but along a family that interpolates between doing nothing and doing all of it, take the Jacobian of each infinitesimal step, and integrate. The result is the sum of a chiral and a Weyl anomaly term, each a trace that has to be regularised — the same heat-kernel regularisation Fujikawa uses, run here with the spin field still in the Dirac operator.

For physicists

with over the Hilbert space and the spinor indices, regularised as . Two features are not standard. The generators and themselves depend on , because the transformation is being built up rather than applied once; and the quartically divergent term in , normally discarded, has to be kept here, since it is multiplied by before the integral.

Eqs. (22)–(25)

Assembled for a texture written as a constant plus a fluctuation, and with the transformation kept to its linear solution, the anomalous action has four terms. The spins started out with whatever action they had; these are what the fermions gave them.

A coupling to the electromagnetic field.

The chiral-anomaly term of a Weyl semimetal, now carrying the fluctuation as well as the mean: a coupling of the spins to , mediated by the fermions. It is this term that carries the transport.

The same shape, built out of spins alone.

Three spins where the first term had a spin and two gauge fields. A three-spin term of this kind is what one expects from the Wess–Zumino term in the low-energy action of fermions coupled to local moments.

A kinetic term the spins did not start with.

, generated entirely by the coupling to the itinerant fermions.

A long-range interaction between the spins.

— the RKKY interaction, with a coupling that depends on the cutoff and grows under the renormalization group, .

The cutoff dependence is not a defect of the method. It is the vacuum polarisation divergence of quantum electrodynamics, governed by the same set of diagrams. In a solid, departure from a linear dispersion cuts it off and leaves a finite but non-universal coupling.

What comes out at the far end

The integral in that action is forbidding and the paper does not attempt it — the four terms above are its linearised form. The transport does not need it. A current is a variation with respect to the gauge field at , where the chiral generator is exactly and the Weyl generator vanishes. So the currents come out exactly, at the order to which the transformation was solved.

For physicists

and a density . In Fourier space each is a convolution of the applied field with , the expectation taken in the effective spin action above — or in whatever mean-field configuration is imposed.

Eqs. (29)–(31)

The first term is the Hall current of a Weyl semimetal, generalised to a texture that varies. The second is a chiral magnetic effect, and it is the one worth pausing on. In the simple Weyl case a chiral magnetic effect requires , which means broken inversion symmetry. Here , the symmetry is not broken, and there is a chiral magnetic effect anyway — generated by the fact that the transformation depends on time.

texture turning at held still peak ∇φ — peak ∂ₜφ —

The phase does the work. Left, a spin texture. Middle, the chiral phase it induces — solved here from , which is what becomes when retardation is dropped and the texture is taken uniform along the third direction — with the Hall current drawn on it for out of the page. Right, the chiral magnetic term: out of the page in bronze, into it in blue. A constant texture gives a uniform phase gradient and nothing else, which is the textbook Weyl case. Turn the texture and the third panel switches on, with no inversion breaking anywhere. is exactly proportional to the rate, so that slider changes the size of the third panel and nothing else about it.

The method also gives correlation functions. The fermion propagator factorises into a free part and a bosonic one — the hallmark of bosonization, and in one dimension the shortest route to non-Fermi-liquid behaviour.

The anomaly used here as a tool is the same one measured as an effect in Chiral Anomaly in Interacting Condensed Matter Systems, and the method is taken further — through gravity, spin–spin interactions, and modes the anomaly generates on its own — in Path-Integral Approach to Quantum Anomalies in Interacting Models.