## Topological magnetoelectric effect versus quantum Faraday effect

1. Observation of topological Faraday and Kerr rotations in quantum anomalous Hall state by terahertz magneto-optics.

Authors: K.N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K.S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura

arXiv:1603.02113

2. Quantized Faraday and Kerr rotation and axion electrodynamics of the surface states of three-dimensional topological insulators.

Authors: L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N.P. Armitage.

arXiv:1603.04317

3. Observation of the universal magnetoelectric effect in a 3D topological insulator.

Authors: V. Dziom, A. Shuvaev, A. Pimenov, G.V. Astakhov, C. Ames, K. Bendias, J. Böttcher, G. Tkachov, E.M. Hankiewicz, C. Brüne, H. Buhmann, and L.W. Molenkamp.

arXiv:1603.05482

Recommended with a commentary by Carlo Beenakker, Leiden University.

|View Commentary|

JCCM_April_2016_01

We thank Prof. Beenakker for the commentary and his interest in this subject. It is indeed important to emphasize that what is measured in all three experiments is the sum of the Hall conductivity of the top and bottom surfaces. And as far as phenomenology is concerned, it can be important to distinguish an explicitly topological magnetoelectric effect (TME) from (as Beenakker calls it) the quantum Faraday effect (QFE). These considerations necessitate certain caveats when thinking of TIs not as surface conductors, but as bulk magnetoelectrics. As we wrote, “Note that unlike the typical case of a magnetoelectric which must break both TRS and inversion symmetry, a TI in magnetic field breaks only TRS. This means that the magnetoelectric effect can be considered as such only locally for a particular surface as the surface trivially breaks inversion. The net effect of considering a local TRS breaking field that is oppositely directed for the two surfaces in our thin film geometry gives Eq. 3.”. Here TRS is time reversal symmetry and Eq. 3 is the expression for the QFE on a substrate.

A few comments are in order. First, it is important to emphasize that both the QFE and TME arise through the E.B term added to the usual Lagrangian and as such they are simply different (partial) manifestations of the same physics. Moreover, although it is correct to say that both the QFE and TME arise via a Hall effect in the surface, we don’t believe it implies that it is JUST a quantum Hall effect in the surface. In our case, in fact the dc quantum Hall effect is not well-developed in the field range that we probe with THz. This presumably occurs because in a Hall bar geometry the side surfaces see mostly a sideways field and are not gapped. A combination of surface disorder and very high fields is required to localize these states as shown by our Hall effect measurements. The fact that the dc Hall effect is not well-developed, but the ac effect is, shows the utility of thinking of TIs, not as surface conductors, but as bulk magnetoelectrics. However this must be done with the caveat that the TME can only be considered as a magneto-electric effect for one surface locally.

3D topological insulators and their quantized theta term requires TRS and so there are some subtleties when defining their response functions under a TRS breaking field. In the original proposal of Qi et al., the quantized Faraday rotation angle was to be defined in the limit of vanishing (but finite) TRS breaking field. In addition to the work detailed in these papers, it would also be important to measure the difference in the Hall conductivities on the two surfaces. Among other things, this would presumably reveal the sensitivity of the quantization to the non-negligible time-reversal symmetry violating fields that are required in real experiments. An explicit example of how this might occur can be found in Everschor-Sitte, Sitte, and MacDonald PRB 2015.

Liang Wu

N. P. Armitage

for their co-authors.