Orbital Altermagnetism

Abstract

We introduce the concept of \emph{orbital altermagnetism}, a symmetry-protected magnetic order of pure orbital degrees of freedom. It is characterized with ordered anti-parallel orbital magnetic moments in real space but momentum-dependent orbital band splittings, analogous to spin altermagnetism. Using a minimal tight-binding model with complex hoppings in a square-kagome lattice, we show that such order inherently arises from staggered loop currents, producing a $d$-wave-like orbital-momentum locking. First-principles calculations show that orbital altermagnetism emerges independent of spin ordering in in-plane ferromagnets of CuBr$_2$ and VS$_2$, so that it can be unambiguously identified experimentally. On the other hand, it may also coexist with spin altermagnetism, such as in monolayer MoO and CrO. The orbital altermagnetism offers an alternative platform for symmetry-driven magnetotransport and orbital-based spintronics, as exemplified by large nonlinear current-induced orbital magnetization.

Figures (9)

• Figure 1: (a) Unit cell (dashed box) of the tight-binding model described by Eq. (\ref{['eq:ham']}) includes three types of sites labeled A, B, and C in a square-kagome lattice, respectively. The nearest-neighbor hopping $t_2e^{i\phi}$ has a complex phase (red arrow), and next-nearest-neighbor hopping $t_1$ (blue) is real. Arrows indicate the direction of positive phase hopping. (b) Band structure of the model. (c) ${\bm k}$-space distribution of the orbital angular momentum $L_z$ for the fourth band, with $k_x$ and $k_y$ in reduced coordinates. (d) Loop-current pattern in the model. Arrows indicate the calculated inter-site currents ($\sim5.4\, \mu$A), with red (+) and blue (-) regions denoting positive and negative $z$-components of the orbital magnetic moment $M_z$. (e) Real-space distribution of orbital magnetic moments in a $10\times 10$ lattice under open boundary conditions at $\mu=0$ eV. Red (blue) indicates a positive (negative) $M_z({\bm r})$. (f) Inter-site current as a function of phase $\phi$. Model paramters: $\epsilon_A=-1.0$, $\epsilon_{B/C}=0$, $t_1=-0.6$, $t_2=-1.0$, and $\phi=\pi/3$.
• Figure 2: (a) Top and side views of the monolayer CuBr$_2$. Spin magnetic moments localized on Cu atoms are annotated with red arrows. (b) Band structures of CuBr$_2$, with the color denoting the magnitude of $L_z$ and $S_y$ for each band. The inset displays the first Brillouin zone and high-symmetry points, with $S_1=(0.5,0.5)$ and $S_2=(0.5,-0.5)$. (c) Real-space distribution of orbital magnetic moments in a $10\times10$ lattice under open boundary conditions at $\mu=0$ eV. Red (blue) denotes positive (negative) $M_z({\bm r})$. (d) Momentum-space distribution of $L_z({\bm k})$ for the lowest conduction band (the 43rd band) across the Brillouin zone.
• Figure 3: (a) Top and side views of the monolayer MoO lattice. Green arrows indicate the spin magnetic moments on Mo atoms. (b) Spin-($S_z$) and Orbital- ($L_z$) resolved band structures, where nearly degenerated band crossing points at $\mu=0.133$ eV are marked as $W_1$ and $W_2$. The inset displays the Brillouin zone with high-symmetry points. (c) Momentum-space distribution of $S_z({\bm k})$ (top panel) and $L_z({\bm k})$ (bottom panel) for the highest valence band across the Brillouin zone, with $k_x$ and $k_y$ in reduced coordinates. (d) Real-space distribution of orbital magnetic moments in a $10\times 10$ lattice under open boundary conditions at $\mu$=0 eV. Red (blue) denotes positive (negative) $M_z({\bm r})$. (e) Nonlinear responses $\chi^X_{zyy}$ ($X=L$ or $S$) as a function of chemical potential $\mu$ from first-principles calculations. (f) Momentum-space distribution of $\chi_{zyy}({\bm k})$ across the Brillouin zone at $\mu=0.133$ eV.
• Figure 4: A schematic diagram illustrates how an in-plane ferromagnet can be used to achieve an out-of-plane orbital antiferromagnet. A combined $\{C_n|t\}\mathcal{T}$ symmetry links to sites with opposite orbital angular momentum while maintaining the in-plane spin. Here, the arrangement of other non-magnetic elements is omitted, though they are typically required for the breaking of the $\mathcal{PT}$ or $\tau\mathcal{T}$ symmetry.
• Figure 5: (a) $\chi_{zyy}$ versus the chemical potential $\mu$ for monolayer MoO based on the first-principle calculations. (b) The distribution of $\chi_{zyy}({\bm k})$ across the Brillouin zone, with the $k_x$ and $k_y$ expressed in reduced coordinates. (c) The distribution of $\chi_{zyy}({\bm k})$ near the $W_1$ and $W_2$, where $\mu$ is fixed at 0.133 eV.