Model Hamiltonian for Altermagnetic Topological Insulators

TL;DR

The paper develops a general framework to realize altermagnetic topological insulators by enforcing combined rotational and time-reversal symmetries, $\mathcal{C}_{l}\mathbb{T}$, to obtain momentum-dependent spin splitting without net magnetization. It constructs explicit 2D and 3D tight-binding models with BHZ-like spin-up sectors paired to symmetry-related spin-down sectors, analyzes spin textures, and computes spin Chern numbers on high-symmetry planes to classify phases. The authors employ Magnetic Equivariant K-theory and Mayer–Vietoris sequences to connect bulk invariants to a $\mathbb{Z}/2$ torsion structure and demonstrate how boundary states (surfaces, hinges, corners) reflect the underlying symmetry-protected topology. They propose FeSe monolayers as realistic 2D platforms exhibiting $d$-wave altermagnetic textures with quantized spin transport, suggesting potential routes for spintronic applications without net magnetization.

Abstract

We present models of topological insulating Hamiltonians exhibiting intrinsic altermagnetic features, protected by combined three-fold or four-fold rotational symmetries with time-reversal. We demonstrate that the spin Chern number serves as a robust topological invariant in two-dimensional systems, while for three-dimensional structures, the topological nature is characterized by the spin Chern numbers computed on the $k_z$=$0$ and $k_z$=$π$ planes. The resulting phases support symmetry-protected boundary modes, including corner, hinges and surface states, whose structure is determined by the magnetic symmetry and the local magnetic moments. Our findings bridge the fields of altermagnetism and topological quantum matter, and establish a theoretical framework for engineering spintronic topological systems without net magnetization.

Figures (6)

• Figure 1: (a) Schematic representation of a two-dimensional $\mathcal{C}_{4z}\mathbb{T}$-symmetric altermagnetic system on an square lattice. The diagram illustrates the tight-binding model and hopping parameters, where arrows indicate the hopping interactions. The mass term $M_0$ captures the sublattice asymmetry arising from oppositely aligned local magnetizations. $K_1$ and $K_2$ denote the intra-site hoppings between orbitals ($s$–$s$ and $p$–$p$) along the $x$- and $y$-directions, respectively. $G_1$ and $G_2$ represent the inter-site hoppings (between $s$ and $p$ orbitals) along $x$ and $y$, respectively, as defined in the Hamiltonian in Eqn. \ref{['C4THamiltonian']}. (b) Spin-resolved band structure for the altermagnetic model, with parameters set to $M_0=1$, $K_1$= $2.125$, $K_2$=$\tfrac{1}{K_1}$, $G_1 = 1.75$, and $G_2$=$\tfrac{1}{G_1}$. All parameters are given in $eV$. Time-reversal symmetry breaking leads to opposite spin splitting of the bands along the $\Gamma$–X and $\Gamma$–Y paths.
• Figure 2: Hamiltonian $H_4$ of Eqn. \ref{['C4THamiltonian']} fixing constants $K_1$=$G_1$=$1.01$ with $K_2$=$\tfrac{1}{K_1}$, $G_2$=$\tfrac{1}{G_1}$ and varying $M_0$. a) is the spin$_z$ Chern number with respect to $M_0$, b) are the edge states ($|\psi_n(r)|^2$) for $M_0 \in \{0.2,1,1.8,2.4 \}$ and c) is the spin$_z$ texture of the upper valence band for the four values of $M_0$ (it is the same texture for the four cases). The Hamiltonian models a TI for $0 < |M_0 | < 2$, as the spin Chern number evidences, and the local charge density of the first three phases exhibit the localization on the boundary.
• Figure 3: (a) Phase diagram of the $\mathcal{C}_{3z}\mathbb{T}$-symmetric altermagnetic model described by Eqn. \ref{['C3THamiltonian']}, showing the SCN as a function of the mass term $M_0$. A topological phase transition occurs near $M_0$=$2$. (b) Spin-projected states of the top valence band across the Brillouin zone, and (c) Spin-resolved edge states for a finite system, calculated at $M_0=0.2$, $1.0$, $1.8$, and $2.4$, with fixed parameters $A_0$=$1.01$, $B_0$=$2/3$ and $J_0$=$0.01$. The spin textures exhibit an $f$-wave AM type characteristic of the $\mathcal{C}_{3z}\mathbb{T}$ symmetry, with a nodal ring whose radius decreases as $M_0$ increases. Both the spin textures and edge-state distributions manifest the $\mathcal{C}_{3z}\mathbb{T}$ symmetry of the Hamiltonian in Eqn. \ref{['C3THamiltonian']}.
• Figure 4: $\mathcal{H}_4({\bf{k}})$ Hamiltonian of Eqn. \ref{['3D_H4_Hamiltonian']} with $K_1$=$1.01$, $G_1$=$1.01$, $D_0$=$0.1$, a) $M_0$=$2.1$ and b) $M_0$=$0.5$. The first graph is the SCN of the valence bands on planes perpendicular to the $k_z$-axis. The second is the spin$_z$ texture of the upper valence band on the BZ. The third are the conducting boundary states. a) is a strong TI because of the change of SCN mod 2. The SCN changes by 1 where the spin Weyl point is located, on the $k_z$-axis with $(k_x,k_y)$=$(0,0)$. The altermagnetic feature is almost constant in all value of $k_z$. The edge states are localized on the faces parallel to the $z$-axis. b) is a weak TI because the SCN mod 2 is always 1. The SCN changes from 1 to -1 where the spin Weyl points are located, one on the $k_z$-axis with $(k_x,k_y)$=$(0,\pi)$, and the other of the $k_z$-axis with $(k_x,k_y)$=$(\pi,0)$,. The spin texture flips on the $k_z$ locations of the spin Weyl points. The boundary states are localized in the corners, thus inducing a HOTI phase. The boundary state calculations were performed on a 13$\times$13$\times$13 finite cubic geometry.
• Figure 5: $\mathcal{H}_3({\bf{k}})$ Hamiltonian of Eqn. \ref{['C3T-3D-Hamiltonian']} with constants $B_0$=$2/3$, $J_0$=$0.01$, $A_0$=$1.01$, $D_0$=$0.1$, a) $M_0$=$2.1$ and b) $M_0$=$0.5$. The first graph is the SCN of the valence bands on planes perpendicular to the $k_z$-axis. The second is the spin$_z$ texture of the upper valence band on the BZ. The third are the conducting boundary states. a) is a strong TI because of the change of SCN mod 2. The SCN changes by 1 where the spin Weyl point is located, on the $k_z$-axis with $(k_x,k_y)$=$(0,0)$. The altermagnetic feature is similar for all value of $k_z$, except when the presence of a nodal sphere makes the the spin texture to change of sign. The edge states are localized on the faces parallel to the $z$-axis. b) is a weak TI because the SCN is always 1. There is no change of SCN since there are no spin Weyl points. The spin texture flips sign from $k_z$=$\pi$ to $k_z$=$0$ due to the nodal sphere present on the whole BZ. The boundary states are localized in the hinges, thus inducing a HOTI phase. Boundary states were computed using a finite cubic system of size 13$\times$13$\times$13.