Odd-parity altermagnetism through sublattice currents: From Haldane-Hubbard model to general bipartite lattices

Abstract

We propose the sublattice currents as a feasible route to odd-parity altermagnetism (ALM), where nonrelativistic collinear spin splitting occurs in the bands as an odd function of momentum. In contrast to previously classified ALMs, the sublattice currents break the time-reversal symmetry in the nonmagnetic crystal structure and allow for such odd-parity spin splitting. A representative example is the Haldane-Hubbard model at half filling. Although the compensated collinear magnetic ground state was previously recognized as antiferromagnetism, we show that sublattice currents induce spin splitting in the bands and therefore turn it into an odd-parity ALM. Interestingly, its topological version serves as an example of ALM Chern insulator. We further generalize the Haldane-Hubbard model to common two- and three-dimensional bipartite lattices. With spin splitting from sublattice currents, the compensated collinear magnetic ground states at half filling are generally odd-parity ALM.

Figures (4)

• Figure 1: Schematic representation of the [$C_2\mathcal{T}||\mathcal{T}$] symmetry in a spinful system without (a) and with (b) time-reversal symmetry breaking in real space, as in the case of e.g. orbital currents (orange arrow). When [$C_2\mathcal{T}||\mathcal{T}$] is preserved, the spin splitting is even in momentum. Red and blue arrows denote up and down spins, respectively. The green font shows what symmetry operation is applied.
• Figure 2: Haldane-Hubbard model and repulsion-driven odd-parity ALM. (a) Haldane model on the honeycomb lattice (gray), where opposite currents flow on the second-neighbor bonds (lighter green and pink) in the two sublattices (green and pink). (b) The repulsion-driven ground state at half filling, where opposite FMs develop on the two sublattices. Here we show the onsite spin orders with colors representing $z$ components. (c) Haldane-Hubbard model with $t_2=0.1$. (i) (Left) Sublattice imbalance $w_{n\mathbf k}$ occurs in the noninteracting bands. (Right) BZ map of total sublattice imbalance $w_{\mathbf k}$ in the occupied bands. (ii) Spin splitting in the ALM at $U_0=4$. (left) The bands split with nonzero spin polarization $s_{n\mathbf k}$. (Right) BZ map of the spin-splitting energy in the occupied bands $E^s_{\mathbf k}$. (iii) BZ map of Berry curvatures in the ALM. (d) Haldane-Hubbard model with $t_2=0.3$ and $U_0=4.8$. The colors in (b)-(d) represent the respective data and follow the respective colorbars.
• Figure 3: 2D bipartite-lattice models and repulsion-driven odd-parity ALMs. For (a) checkerboard and (b) honeycomb lattices, we show (i) model structures, (ii) sublattice imbalances in the noninteracting bands, (iii) opposite FMs in the two sublattices, and (iv) ALM-induced spin splittings in the bands. Note that (b) the honeycomb-lattice model exhibits a different sublattice-current configuration $(1++,1+-,1--)$ from the Haldane model. The formats of the figures follow Fig. \ref{['fig:haldane']}.
• Figure 4: 3D bipartite-lattice models and repulsion-driven odd-parity ALMs. For (a) BCC, (b) 3D checkerboard, and (c) diamond lattices, we show (i) model structures, (ii) sublattice imbalances in the noninteracting bands, (iii) BZs, (iv) opposite FMs in the two sublattices, and (v) ALM-induced spin splittings in the bands. For (b) 3D checkerboard and (c) diamond lattices, we show the results for the sublattice-current configuration $(2++-,2+--)$. For clear illustration of (i) the model structures, we show the sites and currents only in one sublattice. The formats of the figures follow Fig. \ref{['fig:haldane']}.