Zoology of Altermagnetic-type Non-collinear Magnets on the Maple Leaf Lattice

Abstract

We define unconventional non-collinear magnetic ground states on the maple leaf lattice (MLL) distinguished by the selective breaking or preservation of time reversal ($\mathcal{T}$) and parity ($\mathcal{P}$). Depending on the nature of $\mathcal{P}\mathcal{T}$-breaking, linear spin-wave theory reveals momentum-dependent non-relativistic magnon spin splitting at different high symmetry points in the Brillouin zone. From a mean-field analysis of the Hubbard model at weak coupling, we reveal itinerant $\mathcal{P}$-preserving $q=0$ altermagnetic (A$l$M)-type order, while we expect $\mathcal{P}$-broken canted-$120^\circ$ A$l$M-type order at strong coupling. Our findings establish the MLL as a prime platform for exploring phase transitions and frustration phenomena emanating from competing non-collinear A$l$M-type orders.

Figures (3)

• Figure 1: (a) The maple leaf lattice (MLL) exhibits three types of nearest neighbor bonds, $h$, $t$, and $d$, and a subset of second neighbor bonds, $d'$. The bonds correspond to exchange couplings $J_{r}$ or tight-binding hopping amplitudes $t_{r}$, with $r \in \{h,t,d,d'\}$ (lattice vectors indicated). (b)-(d) Three distinct non-collinear, coplanar AlM orders: (b) Canted-$120^\circ$; (c) $q=0$; (d) $q = M$.
• Figure 2: Spin-wave spectrum of (a) canted-$120^\circ$ order for $(J_h,J_t,J_d)=(1.0,1.0,1.5)$ and (b) $q=0$ order for $(J_h,J_t,J_d,J_{d'})=(0.4,0.6,-1.0,-0.25)$. (c), (d) $\langle S^\parallel\rangle$ along constant-energy contours [dashed line in panels (a) and (b)] for the lowest band of the canted-$120^\circ$ (c),(d) Spin splitting around $K, K'$ in (a) and $\Gamma$ in (b). Orders (a) and (b) show parity-even spin momentum singatures.
• Figure 3: Band structure of \ref{['eq:FH']} with $(t_h, t_t, t_d) = (1, 1, 0.5)$ for (a) $U=0$, (b) $U=3$ (altermagnetic metal) and (c) $U=5$ (altermagnetic insulator), plotted along the high-symmetry paths in the BZ. The gray dashed lines mark the Fermi level at half filling. (a) The dot sizes indicate the projection of the Bloch states onto the $d$-orbitals shown in panels (d) and (e) (the numbers specify the relative strength of the sublattice contributions). (b),(c) Double-headed arrows indicate the rigid splitting of the partially flat bands for an altermagnetic metal and insulator. (f) Sublattice resolution of the bands crossing the Fermi level for $U=0$ in \ref{['eq:FH']}. The corresponding Bloch states are predominantly localized on sublattices at opposite corners of the hexamers, indicated by the same colors in panel (g). (h) Expectation values $\langle c_{i,\vec{m}}^\dagger c_{j,\vec{m}} + c_{j,\vec{m}}^\dagger c_{i,\vec{m}} \rangle_0$, on NN bonds for $U=5$. $\vec{m}$ denotes the three constituent magnetic moments, with only the dominant contributions shown.