Following several proposals to accomplish the quantum anomalous Hall eect in topologi- cal insulators [1,2], it has recently been observed in magnetically doped (Bi,Sb)2Te3 , and the exact quantization and edge channel transport have been veried. Shortly after that the tremendous perspectives for metrological applications have been recognized, and recently we have demonstrated in our collaboration with the Physikalisch-Technische Bundesanstalt (PTB), the world record accuracy of the Hall resistance quantization in magnetic topological insulators . Deviation of the Hall resistance from the von-Klitzing constant at zero external magnetic eld in the rst metrologically comprehensive experiment has been already demonstrated to be as low 0:17 0:25 ppm , giving a legitimate hope for a future successful combination of the quantum resistance standard with the Josephson quantum voltage standard in a single measure- ment setup. Such combination could provide a universal quantum standard device capable of providing values of all units based on the Planck constant and the elementary charge.
The mechanism invoked to explain the physics behind the quantum anomalous Hall eect in its initial manifestation is applicable to a 2D system , where the inversion of one spin species is lifted by exchange interaction. The theoretical perspective on the quantum anomalous Hall eect in a three-dimensional topological insulator, however, is dierent, and ties to the axion term characterizing the electrodynamic response of the three-dimensional topological insulator bulk . Due to the eect of the axionic action S = 4 R ~E ~B d3xdt, with the ne structure constant and =1 in the topological insulator bulk up to its boundary, as the magnetic dopants induce a gap, a half-integer contribution to the Hall conductivity xy = 1 2e2=h is expected to be observable as long as the Fermi level resides in the three-dimensional topological insulator bulk gap . A three-dimensional magnetic topological insulator exhibiting the quantum anomalous Hall eect can thus be regarded as an \axion insulator".
Here we present the scaling behavior of V-doped (Bi,Sb)2Te3 samples in the quantum anoma- lous Hall regime. While previous quantum anomalous Hall measurements showed the same scal- ing as expected from a two-dimensional integer quantum Hall state, we observe a dimensional crossover to three spatial dimensions as a function of layer thickness. In the limit of a suciently thick layer, we nd scaling behavior matching the ow diagram of two parallel conducting topo- logical surface states of a three-dimensional topological insulator each featuring a fractional shift of 1 2e2=h in the ow diagram Hall conductivity, while we recover the expected integer quantum Hall behavior for thinner layers. This constitutes the observation of a distinct type of quantum anomalous Hall eect, resulting from 1 2e2=h Hall conductance quantization of three-dimensional topological insulator surface states, in an experiment which does not require decomposition of the signal to separate the contribution of two surfaces. This provides a possible experimental link between quantum Hall physics and axion electrodynamics. 
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