# How is Berry phase connected with chiral anomaly?

Recently I've read in one article about very strange way to describe chiral anomaly on quasiclassical level (i.e., on the level of Boltzmann equation and distribution function).

Starting from Weyl hamiltonian $$H= \sigma \cdot \mathbf p,$$ which describes massless chiral fermions, performing unitary transformation $$|\psi\rangle \to V|\psi\rangle,$$ where $|\psi\rangle$ is fermion state and $V$ is $2\times 2$ matrix that diagonalizes $\sigma \cdot \mathbf p$, such that $$V\sigma \cdot \mathbf pV^{\dagger} = |\mathbf p|\sigma_{3},$$ we obtain the expression for the matrix element $$\langle f|e^{iH(t_{f}-t_{i})}|i\rangle \equiv \left(V_{\mathbf p_{f}}\int Dx Dp \text{exp}\left[i\int dt(\mathbf p \cdot \mathbf x - |\mathbf p|\sigma_{3}-\hat{\mathbf{a}}\cdot \dot{\mathbf p})\right]V_{\mathbf p_{i}}^{\dagger}\right)_{fi}, \quad \hat{a}_{\mathbf p} = V_{p}\nabla_{\mathbf p}V^{\dagger}_{p}$$ By choosing $+1$ helicity and neglecting off-diagonal components of $\hat{a}_{\mathbf p}$ (which is called adiabaticity approximation; it is not valid near $\mathbf p = 0$), we obtain following quasiclassical action: $$S = \int dt (\mathbf p \cdot \dot{\mathbf x} - |\mathbf p| - \mathbf a \dot{\mathbf p})$$ The quantity $\mathbf a$ (which is called Berry phase) plays the role of gauge field in momentum representation, with curvature $$\mathbf b = \nabla \times \mathbf a = \frac{\mathbf p}{|\mathbf p|^{3}}$$ Effect of this Berry phase is absent when $\dot{\mathbf p} = 0$.

If we, however, turn on external EM field, it becomes to be relevant. We obtain that the invariant phase volume element is $$dV = \frac{d^{3}\mathbf x d^{3}\mathbf p}{(2\pi)^{3}}\Omega (\mathbf p), \quad \Omega (\mathbf p) = (1 + \mathbf b \cdot \mathbf B)^{2},$$ where $\mathbf B$ is magnetic field. This captures chiral anomaly effect, $$\tag 1 \dot{\rho} + \nabla_{\mathbf r}(\rho \dot{\mathbf r}) + \nabla_{\mathbf p}(\rho \dot{\mathbf p}) = 2\pi \mathbf E \cdot \mathbf B \delta^{3}(\mathbf p), \quad \rho = f\Omega ,$$ with $\mathbf E$ being electric field and $f$ being distribution function.

I don't understand how this berry phase leads to description of chiral anomaly. They appear due to different reasons (anomaly arises because of non-trivial jacobian of chiral transformation, while berry phase arises because of formal manipulations), anomaly has topological nature connected with difference of number of zero modes of Dirac operator, while Berry phase hasn't such origin. Finally, the rhs of $(1)$ is non-zero only when adiabaticity approximation is violated (and hence the result isn't valid).

Could someone explain me the reason due to which berry phase somehow describes effects of anomaly on quasiclassical level?

• Saying that the anomaly has a "topological nature " and the Berry phase doesn't just shows that you've studied anomalies, but not Berry phases - the latter possess geometrical/topological natures just as the former! See, for instance, this answer and others by DavidBarMoshe - non-triviality of a bundle is certainly a rather geometrical notion, and indeed connected to the mismatched Dirac modes of the chiral anomaly, for the bundle view of anomalies, see Bertlmann's book. Aug 19, 2016 at 19:13

Even though, it is expressed in very simple single point particle models. This is a very recent understanding of the role of the Berry phase in anomalies and its experimental implications. I recommend the following article by: Dwivedi and Stone, for a clear exposition of these facts.

As very well known, the addition of a term of the $$\int i e \mathbf{A}(x).\dot{\mathbf{x}} dt$$ to a particle Lagrangian makes the particle charged with an electric charge $$e$$. In this case, we interpret the field $$\mathbf{A}(x)$$ as the electromagnetic gauge potential.

If instead, or in addition we add a similar term in which $$\mathbf{A}$$ is a Dirac monopole in the momentum space (It is known that this is the Berry connection of a spinning particle), this term modifies the symplectic potential term of the Lagrangian.

As a result we observe the following two phenomena:

1. For a relativistic massless particle, i.e., whose Hamiltonian is:

$$H(p) = |\mathbb{p}|$$

this term locks the spin to the direction of the momentum thus the full Lagrangian describes a spinning relativistic massless particle such that the coefficient of the magnetic monopole is exactly the particle's helicity.

1. When in addition an electromagnetic interaction is switched on, the electromagnetic current becomes anomalous such that the charge non-conservation is given by the known chiral anomaly formula.

Thus the addition of this term to a relativistic massless particle Lagrangian makes it chiral as well as spinning and the anomaly can be deduced by a direct application of the Noether's theorem.

In a subsequent very recent article by Dwivedi and Stone, they generalize the anomaly computation to an arbitrary dimension. Their construction includes a deeper explanation the underlying mechanism.

First please note that the Abelian/non-Abelian Berry phase are holonomies on line/vector bundles over the phase space. The holonomies over infinitesimal loops are given by the first Chern classes of these bundles.

The anomaly, however (by the index theorem) is given by the integrated Chern character over space-time.

Dwivedi and Stone gave an answer of how the Chern class is promoted into the Chern character in the anomaly equation, which I'll give a qualitative explanation in the following. (The actual computations are quite heavy. Most probably subsequent works will appear which will include simplifications)

First, they show that the Liouville measure of their model is not conserved due to the fact that the monopole gauge field which deforms the symplectic structure is singular. Liouville's theorem relies on the fact that the symplectic structure is smooth.

Secondly, the Noether currents are proportional to the (square root of) the Liouville measure of the model, the reason is that for point particles currents must be proportional to the particle's velocity. However, the velocity is a 1-vector while the currents are 1-forms, thus, they must be obtained as Hodge duals of the velocity vector (which includes the multiplication be the square root of the Liouville measure).

Thus the anomaly in the Liouville measure induces the anomaly in the currents.

Now, there is only one term which has a nonvanishing angular integral over the non-compact momentum space, this is the term in which both the non-Abelian gauge field and the Berry fields appear are to the power of half of the space dimension.

This term is divergent and Dwivedi and Stone show how to regularize it. The imprtant observation is that it is the coefficient of the Chern character corresponding to the space-time dimension giving the expected anomaly.

• I have corrected in several places where it was not written that the particle is massless. In addition I am including a qualitative explanation of how the anomaly is obtained from the Berry phase in topology. Sep 1, 2016 at 16:04
• sorry, but what do You mean by "Liouville measure"? Is it the factor $G$ in the front of invariant phase space element $$d\Gamma =G\frac{d^3xd^3p}{(2\pi)^{3}}?$$ Nov 4, 2016 at 8:32
• @NameYYY In principle yes, however due to the ocuurence of the magnetic monopole in the momentum space and the coupling to the Non-Abelian gauge field, the Liouville measure is modified. The way to get the modified Liouville measure is to compute the symplectic potential by means of canonical quantization as Dwivedi and Stone did. Jun 18, 2017 at 16:07

I have been collaborating with the authors of that paper. In Section 2.6 of my thesis I explained the relations between the full quantum computation of chiral anomaly, the (semi-classical) Nielsen-Ninomiya spectral flow picture and the (almost classical) Berry curvature picture to the best details of my knowledge.

Let me briefly summarize the ideas.

Nielsen and Ninomiya has provided a computation of chiral anomaly in terms of spectral flow, in the context of what is known today as Weyl semimetal. Of course, in standard texts like Peskin and Schroeder, chiral anomaly in particle physics was also taught in terms of spectral flow. The distinction is minor. We know chiral anomalies have IR and UV interpretations. The spectral flow through the zero energy (Weyl node) is an IR interpretation; but the particles "must have gone somewhere", and that is capture by the UV boundary conditions. In the Weyl semimetal, the left and right Weyl nodes become connected in the UV (deep in the valence band). On the other hand, for Weyl fermion in particle physics, we don't know what the actual UV is, of course, but we may picture an infinitely deep Dirac sea, and the UV boundary condition is just that the flow towards infinite negative energy are opposite for left and right Weyl fermions -- this assumption is needed for the conservation of the total U(1) charge. So in the two contexts the difference is only what we say about the UV, but they give essentially the same IR physics.

Once we have the picture of spectral flow, we can make easy connection to the Berry phase computation. In the Berry phase computation, the symplectic 2-form is closed except at the $$p=0$$ Weyl point, which means this point in the momentum space is a sink or source of Liouville measure flow. Of course, classical mechanics breaks down at this point (roughly speaking, the classical computation is good only for $$\partial_x \ll p$$), so what the Berry phase computation actually tells is again a boundary condition -- but now it is not a UV boundary condition, but an IR boundary condition around vicinity of the $$p=0$$ point (where classical mechanics fails): What we have computed is how many particles have flowed into / out of the vicinity of $$p=0$$. One finds the amounts are indeed opposite for left and right Weyl fermions, and the amount is in agreement with the quantum computation of chiral anomaly.

This agreement is expected because the (semi-)classical Berry curvature formalism can be extracted from quantum mechanics to first correction in $$\hbar\partial_x$$ (for instance see the paper you referred to; Section 2.4 of my thesis contains a more detailed derivation in general cases, including background electromagnetic field), which is the order of chiral anomaly $$\partial_x J \sim \hbar \partial_x A \partial_x A$$.
On the other hand, in gravitational field, $$\partial_x J \sim \hbar (\hbar\partial_x^2 g) (\hbar\partial_x^2 g)$$, there is no consistent smallness counting that makes it work. There is some counting that allows one to derive the gravitational chiral anomaly if the curvature is purely spatial, but at the sacrifice of Lorentz invariance (since this is kind of unsatisfactory, this is not written up anywhere).
A similar issue exists for non-abelian gauge field, where $$\hbar F\sim \hbar\partial_x A + A^2$$. For consistency of power counting, one might count $$A \sim \hbar\partial_x$$ (an issue that did not exist in the abelian case), hence the non-abelian chiral anomaly exhibits the same issue as the gravitational one. For instance in this paper mentioned in the other answer, they obtained the non-abelian chiral anomaly in a non-Lorentz invariant model and find the correct result; but there is no modification of the semi-classical model that can make it Lorentz invariant.