Orbital magnetoelectric coupling in band insulators

Magnetoelectric responses are a fundamental characteristic of materials that break time-reversal and inversion symmetries (notably multiferroics) and, remarkably, of “topological insulators” in which those symmetries are unbroken. Previous work has shown how to compute spin and lattice contributions to the magnetoelectric tensor. Here we solve the problem of orbital contributions by computing the frozen-lattice electronic polarization induced by a magnetic field. One part of this response (the “Chern-Simons term”) can appear even in time-reversal-symmetric materials and has been previously shown to be quantized in topological insulators. In general materials there are additional orbital contributions to all parts of the magnetoelectric tensor; these vanish in topological insulators by symmetry and also vanish in several simplified models without time reversal and inversion whose magnetoelectric couplings were studied before. We give two derivations of the response formula, one based on a uniform magnetic field and one based on extrapolation of a long-wavelength magnetic field, and discuss some of the consequences of this formula.

Figure 1

A supercell admits a small magnetic flux and the quantum of polarization transverse to the long direction is correspondingly small but the quantum for polarization along the long direction is much larger.

Figure 2

Schematic band structure that leads to vanishing ${\alpha }_G$. The bands below the chemical potential are degenerate with energy $E_{\mathbf{k}}^{\text{v}}$ while the bands above the chemical potential have energy $E_{\mathbf{k}}^{\text{c}}=\text{const}-E_{\mathbf{k}}^{\text{v}}$.

Figure 3

A tetrahedral tight-binding molecule for spinless electrons, with one orbital per site and complex hoppings. The hopping integrals are all equal, except that those around one face have a phase of $i$ relative to the other three. There are then two pairs of degenerate levels.

Figure 4

As outlined in the text, (a) turning on a magnetic field produces a macroscopic polarization through the flow of surface currents while (b) varying the crystal Hamiltonian in the presence of a fixed magnetic field produces a polarization through the flow of current through the bulk.