High spin axion insulator

Abstract

Axion insulators possess a quantized axion field $θ=π$ protected by combined lattice and time-reversal symmetry, holding great potential for device applications in layertronics and quantum computing. Here, we propose a high-spin axion insulator (HSAI) defined in large spin-$s$ representation, which maintains the same inherent symmetry but possesses a notable axion field $θ=(s+1/2)^2π$. Such distinct axion field is confirmed independently by the direct calculation of the axion term using hybrid Wannier functions, layer-resolved Chern numbers, as well as the topological magneto-electric effect. We show that the guaranteed gapless quasi-particle excitation is absent at the boundary of the HSAI despite its integer surface Chern number, hinting an unusual quantum anomaly violating the conventional bulk-boundary correspondence. Furthermore, we ascertain that the axion field $θ$ can be precisely tuned through an external magnetic field, enabling the manipulation of bonded transport properties. The HSAI proposed here can be experimentally verified in ultra-cold atoms by the quantized non-reciprocal conductance or topological magnetoelectric response. Our work enriches the understanding of axion insulators in condensed matter physics, paving the way for future device applications.

Figures (4)

• Figure 1: Model of the HSAI.a Schematic for the HSAI defined on the $|s,m_z\rangle$ space. The blue arrows represent the electron spin with different magnetic quantum number $m_z$, which takes values ranging from $-s$ to $s$ individually. b Energy spectra of the spin-$3/2$ HSAI along M$\rightarrow\Gamma\rightarrow$R path on a slab of thickness $L_z$ with (red solid lines) and without (blue dashed lines) the magnetic exchange interaction. Here, the black lines refer to bulk bands. Inset: Energy dispersion for the spin-$3/2$ HSAI in the absence of magnetic exchange term near the charge neutral point (solid lines) and the fitting data (markers). c Layer-resolved Chern number $C_z$ and the cumulative Chern number $\tilde{C}_z=\sum_{-L_z/2}^zC_z$ versus the layer index $z$ for the spin-$3/2$ HSAI. The surface Chern numbers $C^{top(bot)}_{surf}$ that summarize the layer-resolved Chern number on the upper (lower) half side is $-2$ ($+2$). d Surface Chern number as a function of the Fermi energy $E_F$ for the spin-$3/2$ HSAI. Here, the thickness of the HSAI slab is $L_z=20$.
• Figure 2: Transport properties of the spin-$3/2$ HSAI.a Schematic current flow in a HSAI. The red arrows denote the quantized helical hinge currents. b Energy spectrum and the average position $\langle z/L_z\rangle$ on the front surface for a HSAI nanowire with $L_y=30$, $L_z=16$. c Spectrum density $A(k_x,E)$ for the front lower hinge as labeled by the blue line in a on the $k_x-E$ plane. Here, the system size is $L_y=30$, $L_z=\infty/2$. The white dashed line represents the Fermi energy $E_F=0.1$. The white stars that mark the intersects between the Fermi energy and the spectrum are the Fermi momenta $k_{F1}$ and $k_{F2}$. d Top and middle panels are the current density $J_x(z)$, current flux $I_x(z)$ and its $z$-averaged flux $\langle I_x(z)\rangle$ versus the layer index $z$ for a semi-infinite system with size $L_y=30$, $L_z=\infty/2$. The blue dots are the fitting data. Bottom panel shows the distribution of the moving averaged current $\langle I_x(z)\rangle_{MA}$ on the front surface with system size $L_y=30$, $L_z=150$. e Bird's eye view (top panel) and high-angle shot (bottom panel) for the six terminal device. f Ensemble-averaged non-reciprocal conductances versus the Fermi energy in the clean limit (W=0), with non-magnetic Anderson disorders of strength $W=1$ and with magnetic Anderson disorders of strength $W_z=0.3$. Here, the system size is $L_x=31$, $L_y=20$, $L_z=21$, and the size of transverse terminals is $10\times10$. g Experimental setup to detect the non-reciprocal conductance. In this setup, terminals 2, 4, 5, and 6 are grounded. The voltage is applied alternatively to terminal 1 or terminal 3. h Corresponding temporal dependent current output with parameters $G_{13}=4.5e^2/h$, $G_{31}=2.5e^2/h$. i$F(\omega)$ as a function of the frequency $\omega$.
• Figure 3: Axion term and topological magneto-electric effect.a and f Hybrid Wannier charge centers $z_{n{\bf{k}}}$ along R$\rightarrow$$\Gamma$$\rightarrow$M$\rightarrow$R loop inside the first Brillouin zone for a six-layer HSAI slab with spin-$3/2$ (a) and spin-$5/2$ (f), respectively. b and g are corresponding axion terms and the surface Chern numbers versus the inverse layer thickness obtained by using the HWFs. c Magnetic field induced charge distribution along $\hat{z}$-direction and the layer-resolved Chern number for a spin-$3/2$ HSAI with $L_z=24$. Here, the charge polarization is obtained on a HSAI slab with open boundary condition along $\hat{y}$-direction ($L_y=40$) but periodic boundary condition along $\hat{x}$-direction. The magnetic flux inside one unit cell is $\phi_0=Ba_0^2=0.01h/e$. d Electric field induced orbital magnetization for a spin-$3/2$ HSAI with $L_z=20$. The black dashed line shows the ideal case (IC) with an exact axion term $\theta=4\pi$. e Size scaling of the axion term $\theta_{CS}^{slab}/\pi$, polarization coefficient $P/(\alpha\phi)$, and magnetization coefficient $M/(\alpha\delta U)$ at $\delta U=0.001$. We have checked that the slight deviation of $P/(\alpha\phi)$ originates from the finite size effect.
• Figure 4: Phase transition in spin-$3/2$ HSAI.a Canted HSAI under an in-plane magnetic field. $\gamma$ is the canting angle between the magnetic vector and $\hat{z}$-axis (polar angle). b Energy gaps versus $\gamma$ for spin-$3/2$ HSAI and for spin-$1/2$ axion insulator, respectively. c Energy spectra for the HSAI with spin-$3/2$ (black solid lines) and spin-$1/2$ (blue dashed lines) at $\gamma=\pi/4$. d Hybrid Wannier charge center $z_{n{\bf{k}}}$ as a function of $k_x$ for a HSAI slab at $\gamma=\pi/4$. e Surface Chern numbers obtained from the effective Hamiltonian in Eq. (\ref{['model_Ham']}) and the HWFs versus $\gamma$. f Axion term $\theta_{CS}^{slab}/\pi$, polarization coefficient $P/(\alpha\phi)$ ($\phi_0=0.01h/e$) and magnetization coefficient $M/(\alpha\delta U)$ ($\delta U=0.001$) versus $\gamma$. The thickness of the HSAI is $L_z=20$.