Three-dimensional (3D) topological insulators (TIs) are unusual quantum materials that host conducting helical Dirac states on their surfaces, which are protected by time reversal symmetry (TRS), but are electrically insulating in the bulk [1,2]. TI is distinct from a trivial insulator by its unique electromagnetic response, described by the so-called  term shown below in addition to the ordinary Maxwell terms [3][4][5][6]. Here E and B are the conventional electric and magnetic fields inside an insulator, e is electron charge, and θ is the dimensionless pseudo-scalar parameter describing the insulator. For a trivial insulator, θ=0, while for a TI, θ=. When TRS is preserved, θ is either 0 or , reflecting its topological nature. This  term is related to the axion electrodynamics in particle physics [7]. Since the EB  term can be rewritten as a total derivative, its effect manifests on the surface states. A half-integer quantum Hall effect on the TI surface occurs once the surface Dirac Fermions acquire a mass, i.e. the surface state is gapped by magnetism. Such half-integer quantum Hall effect on TI surface can lead to a variety of exotic phenomena such as the quantum anomalous Hall (QAH) effect [3,[8][9][10][11][12][13][14][15], the quantized magneto-optical effect [3,16,17], the topological magnetoelectric (TME) effect [3][4][5][6]18], and the image magnetic monopole [19]. The QAH and quantized magneto-optical effects have been experimentally demonstrated in pure or magnetic TI films [10][11][12][13][20][21][22]. The TME effect refers to the quantized response of electric 3 polarization to applied magnetic fields and vice versa. The realization of the TME effect requires the following three conditions: (i) the TI film should be in the 3D regime; (ii) all the surfaces are gapped with the chemical potential lying within the gaps; (iii) the interior of the TI maintains TRS or inversion symmetry to maintain  in the bulk. A material system allowing for the realization of TME effect is known as an axion insulator [3][4][5].Recently, two papers reported the possible realization of the axion insulator. (Figs. 3k and 3m). We note that the zero  yx plateau is absent in either uniformly doped or the Cr modulation doped QAH samples [23,26,31,32]. The 3-5-3 SH2 was magnetically trained first by an upward sweep up to  0 H=1.5T before being swept downward. When  0 H=-0.01T, the MFM contrast is uniform (red), indicating that both top V-and bottom Cr-doped TI layers have upward magnetization (Fig. 3c). At  0 H=-0.05T, some reversed magnetic domains (green regions in Fig. 3d) appear, presumably in the 'softer' Cr-doped TI layer. As  0 H is swept further, the green regions expand and fill up the whole scan area at  0 H=-0.09T, indicating the uniform antiparallel magnetization alignment over the entire 3-5-3 SH2. When  0 H is further swept toward H c1 , new reversed magnetic domains (blue regions in Fig. 3g) nucleate at different locations, presumably in the 'harder' V-doped TI layer. Downward parallel magnetiza...

No abstract