Adiabatic Pumping of Chern-Simons Axion Coupling

We study the adiabatic pumping of the Chern-Simons axion (CSA) coupling along a parametric loop characterized by a nonzero second Chern number $C^{(2)}$ from the viewpoint of the hybrid Wannier representation, in which the Wannier charge centers are visualized as sheets defined over a projected 2D Brillouin zone. We derive a new formula for the CSA coupling, expressing it as an integral involving Berry curvatures and potentials defined on the Wannier charge center sheets. We show that a loop characterized by a nonzero $C^{(2)}$ requires a series of sheet-touching events at which $2\pi$ quanta of Berry curvature are passed from sheet to sheet, in such a way that $e^2/h$ units of CSA coupling are pumped by a lattice vector by the end of the cycle. We illustrate these behaviors via explicit calculations on a model tight-binding Hamiltonian and discuss their implications.

Figure 1

The two WCC sheets of the half filled FKM model, and one set of periodic images, at four stages $\alpha =(0,3\pi /4,\pi ,5\pi /4)$ along the parametric cycle (clockwise from upper left). Blue and red colors show positive and negative values of Berry curvature ${\mathrm{\Omega }}_z$ on the sheets, respectively. The Chern numbers associated with the individual WCC sheets are shown for those cases where sheets do not touch.

Figure 2

WCCs at (a) $M$ and (b) $\mathrm{\Gamma }$ of the 2D BZ for the FKM model as a function of the adiabatic loop parameter $\alpha$. Blue and red colors indicate positive and negative values of Berry curvature, respectively. (c) CSA coupling $\theta (\alpha )$ and its contributions ${\theta }_{z\mathrm{\Omega }}$ and ${\theta }_{\mathrm{\Delta }xy}$ for the same adiabatic loop.