fermions - 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?

Answer

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.

Notes added:

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.