Side-surface-mediated hybridization in axion insulators

The axion insulator is believed to host half-quantized chiral currents running antiparallelly on its top and bottom surfaces. However, the experimental detection of the half quantization in axion insulators remains elusive. In this paper, we propose a mechanism to explain why the half quantization is hard to be probed by showing that the half-quantized counterpropagating currents in axion insulator thin films are strongly suppressed due to the hybridization mediated by the massless side-surface states. This side-surface-mediated hybridization leads to a different type of finite-size effect, which features a power-law decay with the increasing film thickness, different from the exponential decay in topological insulators. Moreover, we show that the half quantization can be extracted in the axion insulator phase by adopting the nonlocal transport measurement.

Figure 1

(a) Schematic illustration of the Chern insulator phase in a magnetic topological insulator. Each of the magnetic top and bottom surfaces hosts a gapped Dirac cone characterized by a half-quantized Hall conductance. (b), (c) Numerically calculated transmission coefficients as functions of (b) the thickness $n_z$ with $E_F=0.01$ and (c) Fermi energy $E_F$ with different film thicknesses $n_z$. The system size is taken as $n_x=n_y=30$. The magnetization strength for the Chern insulator phase is taken as $m_t=m_b=0.15$. Here, $T_1, T_2$, and $T_3$ are the transmission coefficients between two clockwise neighboring, counterclockwise neighboring, and non-neighboring terminals, respectively. (d) Schematic illustration of the four-terminal Hall-bar setup. $T_{1,2,3}$ can be obtained through the nonlocal transport. (e)–(g) The same as (b)–(e) except that they depict the axion insulator phase with $m_t=-m_b=0.15$. The superscripts “Chern” in (b) and “Axion” in (f) are to distinguish the Chern and axion insulator phases, respectively. (h) The transmission coefficients as functions of $m_b$, where we take $m_t=0.15$. The system size is taken as $n_x=n_y=30$ and $n_z=50$.

Figure 2

(a) The spectrum of the axion insulator phase as a function of $k_y$. Here, the color scheme indicates the wave-function distribution shown in (c) and (d). (b) The hybridization energy gap ${\mathrm{\Delta }}_E$ at $k_y=0$ as a function of the film thickness $n_z$. The inset shows ${\mathrm{\Delta }}_E$ exhibits a linear relationship as a function of $1/n_z$. (e) and (f) The probability distributions ${\left|\psi (x=1,z)\right|}^2$ [the area in the dashed frame in (e) and (f)] as functions of $z$ at $k_y=-0.01$ (red), $k_y=0$ (green), and $k_y=0.01$ (blue) with film thickness (e) $n_z=20$ and (f) $n_z=60$, respectively. The results are calculated in the axion insulator phase with the periodic boundary condition along the $y$ direction and the open boundary conditions along the $x$ and $z$ directions. The system sizes are (a), (c)–(e) $n_x=n_z=20$, (b) $n_x=20$, and (f) $n_x=20$ and $n_z=60$, respectively.

Figure 3

(a) The side surface of the axion insulator phase can be described by a massless region sandwiched by two massive regions. (b) The topological insulator can be described by a massive region sandwiched by two massive regions. (c), (d) Probability density of the solutions in the mass domains at $k_x=0$. Here, green, yellow, and white regions are characterized by different topological masses, respectively. (e) The hybridization gap ${\mathrm{\Delta }}_E$ of the solution in (d) as a function of $L$. We take $m=0.2$ in (c)–(e).