Axion response in gapless systems

The strong topological insulator in 3D is expected to realize a quantized magnetoelectric response, the so-called axion response. However, many of the materials predicted to be topological insulators have turned out to be metallic, with bulk Fermi surfaces. Following the result of Bergman and Refael Phys. Rev. B 82, 195417 (2010)] that the surface states of the topological insulator persist even when the band structure gap is closed, we explore the fate of the magnetoelectric response in such systems. We find that a nonquantized magnetoelectric coupling remains once a bulk Fermi surface opens. More generally, we find higher-dimensional analogs of the intrinsic anomalous Hall effect for all Chern forms-quantized transport coefficients in the gapped case become nonquantized when the gap is closed. In particular, the nonquantized magnetoelectric response in 3D descends from the intrinsic anomalous Hall effect analog in 4D.