Magnetic responce in the quantized spin Hall system with electron correlation

We investigate the magnetic response in the quantized spin Hall (SH) phase of layered-honeycomb lattice system with intrinsic spin-orbit coupling $\lambda_{\rm SO}$ and on-site Hubbard $U$. The response is characterized by a parameter $g= 4 U a^2 d / 3$, where $a$ and $d$ are the lattice constant and interlayer distance, respectively. When $g< (\sigma_{xy}^{s2} \mu)^{-1}$, where $\sigma_{xy}^{s}$ is the quantized spin Hall conductivity and $\mu$ is the magnetic permeability, the magnetic field inside the sample oscillates spatially. The oscillation vanishes in the non-interacting limit $U \rightarrow 0$. When $g > (\sigma_{xy}^{s2} \mu)^{-1}$, the system shows perfect diamagnetism, i.e., the Meissner effect occurs. We find that superlattice structure with large $a$ is favorable to see these phenomena. We also point out that, as a result of Zeeman coupling, the topologically-protected helical edge states shows weak diamagnetism which is independent of $g$.