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.

Fig. 1. Model of the HSAI.

a Schematic for the HSAI defined on the $∣s,m_z⟩$ 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 → Γ → 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 ${\tilde{C}}_z={\sum }_{-L_z/2}^z{C}_z$ versus the layer index z for the spin-3/2 HSAI. The surface Chern numbers $C_{surf}^{top\left(bot\right)}$ 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.

Fig. 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 〈z/L_z〉 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 = ∞/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 〈I_x(z)〉 versus the layer index z for a semi-infinite system with size L_y = 30, L_z = ∞/2. The blue dots are the fitting data. Bottom panel shows the distribution of the moving averaged current ${⟨I_x\left(z\right)⟩}_{MA}$ 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 × 10. 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 G₁₃ = 4.5e²/h, G₃₁ = 2.5e²/h. i F(ω) as a function of the frequency ω.

Fig. 3. Axion term and topological magneto-electric effect.

a and (f) Hybrid Wannier charge centers z_nk along R → Γ → M → 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 $\widehat{z}$-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 $\widehat{y}$-direction (L_y = 40) but periodic boundary condition along $\widehat{x}$-direction. The magnetic flux inside one unit cell is ${\varphi }_0=B{a}_0^2=0.01h/e$. 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 θ = 4π. e Size scaling of the axion term ${\theta }_{CS}^{slab}/\pi$, polarization coefficient P/(αϕ), and magnetization coefficient M/(αδU) at δU = 0.001. We have checked that the slight deviation of P/(αϕ) originates from the finite size effect.

Fig. 4. Phase transition in spin-3/2 HSAI.

a Canted HSAI under an in-plane magnetic field. γ is the canting angle between the magnetic vector and $\widehat{z}$-axis (polar angle). b Energy gaps versus γ 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 γ = π/4. d Hybrid Wannier charge center z_nk as a function of k_x for a HSAI slab at γ = π/4. e Surface Chern numbers obtained from the effective Hamiltonian in Eq. (1) and the HWFs versus γ. f Axion term ${\theta }_{CS}^{slab}/\pi$, polarization coefficient P/(αϕ) (ϕ₀ = 0.01h/e) and magnetization coefficient M/(αδU) (δU = 0.001) versus γ. The thickness of the HSAI is L_z = 20.