Altermagnetism and its induced higher-order topology on the Lieb lattice

TL;DR

Altermagnetism on the Lieb lattice is explored through a spin-cluster design that yields $d$-, $d_{xy}$-, and $g$-wave AM configurations and is coupled to spin-orbit interactions. The work shows that AM reconstructs edge-state spectra, opens gaps at Dirac points, and, in open square geometries, yields corner (higher-order) modes via mass-domain-wall mechanisms, with the HOTop phase being particularly pronounced for in-plane AM. Mirror-symmetry and spin-resolved subspaces lead to a $C_{ m M}=1$ mirror Chern character in certain regimes, while in-plane AM can gap edge states and induce robust HOTop across multiple AM symmetries. Overall, AM emerges as a powerful tool to engineer higher-order topological states on the Lieb lattice, with potential implications for material realization and heterostructure design.

Abstract

Altermagnetism (AM) has brought renewed attention to the Lieb lattice. Here, we broaden the scope of altermagnetic models on the Lieb lattice by using a general scheme based on spin clusters. We design various altermagnetic models with d- and g-wave on the Lieb lattice, and investigate its interplay with spin-orbit coupling. While the altermagnetic unit cell reconstructs the topological edge states in the strip geometry and leads to the emergence of Dirac points, the in-plane magnetic moments of AM can induce gaps at these points. In an open square geometry, corner modes emerge within these gaps, realizing higher-order topological states. We further verify that the induction of higher-order topology is applicable to all altermagnetic configurations constructed here on the Lieb lattice, and is most pronounced for AM by comparing with the other types of magnetism such as ferromagnetism and ferrimagnetism. Our results highlight the exotic properties of AM, and suggest its potential applications in engineering topological quantum states.

Figures (10)

• Figure 1: Schematic representation of the process of constructing a Lieb lattice using a three-site cluster. (a) The basis obtained by applying $C_{4z}$ to the three-site cluster. (b) The square Bravais lattice. (c) The Lieb lattice formed by attaching the basis from (a) to the square lattice in (b).
• Figure 2: (a) Two sublattices of the bipartite Lieb lattice. (b)-(f) The constructed altermagnetic models, in which at least one sublattice is fully occupied. (g) A magnetic configurtion exhibitting $g$-wave AM. The thick bonds in (d) and (e) have a different hopping amplitude, $t_2$, resulting from the distortion introduced to break the symmetry that protects the spin degeneracy. All other bonds have a hopping amplitude of $t_1$. The $2\times2$ plaquette enclosed by the orange box is labeled as $(\uparrow,\uparrow,\uparrow,\uparrow)$ [(d) and (e)], and as $(\uparrow,\uparrow,\downarrow,\downarrow)$ [(f)].
• Figure 3: (a) The energy spectra of the Hamiltonian in Eq.(1) along the symmetric lines of the Brillouin zone for $\lambda=0$ (dotted line) and $\lambda=0.4$ (solid line). (b) The Fermi surfaces at $\mu=0.2$ for the AM with $\lambda=0$. The energy spectrum on a strip geometry with the magnetic moment in the $z$ direction (c) and in the $x$ direction (d). (e) Spatial distributions of the spin-up and spin-down edge states at $k_x=\pi/4$, indicated by the cross in (c), showing a monotonic decay with exponential modulation. The red dotted line in (e) represents an exponential fit to the squared amplitudes on the same sublattice. (f) Spin distributions $\langle s_x\rangle,\langle s_z\rangle$ of the edge states at $k_x=\pi/4$ in (d). The in-plane altermagnetism tilts the edge-state spins toward the $x$ direction, making them nonparallel. Here the nearest-neighbor hopping amplitude is $t=1$, and the exchange coupling in (c) and (d) is $J=0.4$.
• Figure 4: (a) Schematic of the Lieb lattice geometry with edges oriented in the $45^{\circ}$ direction. (b) The spectra of the corresponding strip geometry with (red) and without (blue) AM. (c) The spectrum of the open square geometry around the lower gap, showing four corner modes emerging in the gap. (d) The spin-resolved distributions of the corner modes near the Fermi energy. Due to the AM aligned in the $x$-direction, the spin is polarized in the $x$ (red) or $-x$ (blue) direction. (e) and (f) The spectrum around the upper gap and the spin-resolved distributions of the corner modes within the gap. Insets of (d) and (f) show a enlarged view of the corner. The parameters are $t = 1$, $\lambda = 0.4$, and $J=0.1$.
• Figure 5: (a) Schematic of the $d_{xy}$-wave AM in a Lieb-lattice geometry with edges oriented in the normal ($0^{\circ}$) direction. (b) The Fermi surface for the altermagnetic model in (a) with $t = 1,J=0.4,\lambda = 0$, and $\mu=0.9$. (c) The spectrum of the corresponding strip geometry of (a), with (red) and without (blue) AM for comparison. (d) The distributions of the corner modes in the open square geometry of (a). Inset of (d) shows the corresponding spectrum, revealing four corner modes emerging in the gap. The exchange coupling is set to $J = 0.1$ in (c), and $J = 0.4$ in (d). The remaining parameters in (c) and (d) are $t = 1$ and $\lambda = 1$.