Multiple Topological Phases Controlled via Strain in Two-Dimensional Altermagnets

Abstract

Altermagnets (AMs) are an emergent class of magnetic materials that combine properties of ferromagnets and antiferromagnets, exhibiting spin-polarized Fermi surfaces and zero net magnetic moment due to combined time-reversal and crystal symmetry. Here, we construct a Kondo-lattice model on a two-dimensional square Lieb lattice to investigate the topological properties of AMs. We identify a type-II quantum spin Hall state characterized by spin-polarized counterpropagating edge states. Breaking the $C_{4z}\mathcal{T}$ symmetry, which connects magnetic sublattices, induces a transition to a quantum anomalous Hall state. We further establish a strain-induced mechanism to control these topological phase transitions and present the corresponding phase diagram. Finally, we demonstrate the predicted transitions in monolayer CrO, a realistic altermagnetic candidate, using first-principles calculations. Our findings highlight the potential of 2D AMs as a versatile platform for topological spintronics, enabling strain-tunable helical and chiral edge states within a single system.

Figures (4)

• Figure 1: (a) Top view and front view of the lattice structure. Red and blue spheres represent the same magnetic atoms with opposite magnetization, white and gray spheres correspond to different nonmagnetic atoms. (b) Schematic diagram of the first Brillouin zone. (c) Schematic structure of the altermagnet. Red and blue atoms denote magnetic ions with opposite magnetizations, whose onsite potentials are $\mu + u\sigma_z$ and $\mu - u\sigma_z$, respectively. The arrows indicate NN and NNN hoppings, while the counterclockwise NN hopping amplitudes are $t/4 - i\lambda \sigma_z/4$.
• Figure 2: Band structures and Berry curvature for (a) $u=-2.2$ and (c) $u=-1.6$. The other parameters are set as $A=0.5$, $B=-1$, $C=0.5$, $D=1$, $\lambda=0.5$, $\mu=1$, and $t=4$. The details of the Berry curvature near the X valley in the trivial phase are also shown in (a). The local density of states (LDOS) of the corresponding edge states are shown in (b) and (d). Here, Spin-polarized LDOS is obtained by subtracting the spin-down component from the spin-up component: deeper red indicates a larger contribution from spin-up states, while deeper blue corresponds to spin-down states. In (e) and (f), we explicitly break the $C_{4z}\mathcal{T}$ symmetry by adding an additional term $\mu_s\tau_z\sigma_0$($\mu_s=0.6$ and $u=-2.2$), and present the resulting band structure and edge states.
• Figure 3: (a) The phase diagram of the Chern number $\mathcal{C}_{\uparrow}$ for different values of u and B. It can be seen that the phase boundary corresponds to $|u/B|=2$, and the sign of $B$ determines the value of the $\mathcal{C}_{\uparrow}$. (b) Schematic phase diagram under various strain conditions, with the origin denoted by the point O. Red and blue lines indicate $|u_{\uparrow}/B_{\uparrow}|=2$ and $|u_{\downarrow}/B_{\downarrow}|=2$, respectively.
• Figure 4: (a) Phase diagram for different lattice constants. The black line denotes the diagonal, serving as an indicator of biaxial strain. (The data of blue and red line are in SM Part VI). (b) Band structure without SOC at a lattice constant of 3.40Å and 3.30Å (lower panel). Red and blue lines indicate spin-up and spin-down. Dashed lines correspond to the results of DFT calculations, whereas solid lines represent the model fits obtained from our theoretical framework (c) The black curve shows the energy difference at the X valley as a function of the lattice constant, while the red curve indicates the cosine value of the valley position. (d) The blue and black curves represent the fitted values of $u$ and $B$ at different lattice constants, respectively. The red curve shows the magnetic moment of a single magnetic atom as a function of lattice constants.