Flux States and Topological Phases from Spontaneous Time-Reversal Symmetry Breaking in $\mathrm{CrSi}(\mathrm{Ge}){\mathrm{Te}}_3$-Based Systems

We study adatom-covered single layers of ${\mathrm{CrSiTe}}_3$ and ${\mathrm{CrGeTe}}_3$ using first-principles calculations based on hybrid functionals. We find that the insulating ground state of a monolayer of La (Lu) deposited on single-layer ${\mathrm{CrSiTe}}_3$ (${\mathrm{CrGeTe}}_3$) carries spontaneously generated current loops around the Cr sites. These “flux states” induce antiferromagnetically ordered orbital moments on the Cr sites and are also associated with nontrivial topological properties. The calculated Chern numbers for these systems are predicted to be $\pm 1$ even in the absence of spin-orbit coupling, with sizable gaps on the order of 100 meV. The flux states and the associated topological phases result from spontaneous time-reversal symmetry breaking due to the presence of nonlocal Coulomb interactions.

Figure 1

(a) The lattice structures of bulk ${\mathrm{CrSiTe}}_3$ and ${\mathrm{CrGeTe}}_3$. (b),(c) The structure of single-layer ${\mathrm{CrSiTe}}_3$ (${\mathrm{CrGeTe}}_3$) with deposited adatoms, (b) a 3D view, and (c) a top view, with $t_1$ and $t_2$ denoting the 2D primitive lattice vectors, and a sketch of the hexagonal Brillouin zone.

Figure 2

The band structures in the absence of SOC for (a) Ge-Lu and (b) Si-La. The red and blue curves represent the spin-minority and spin-majority bands, respectively. The band structures are plotted along the high-symmetry path marked by magenta lines in the Brillouin zone shown in Fig. 1.

Figure 3

Current loops flowing between the Te atoms in the majority-spin channel of the SOC-free Ge-Lu system are denoted by black arrows, in units of $\mu \mathrm{A}$. The magenta arrows denote orbital magnetic moments on the Cr sites.

Figure 4

Left: Edge-state spectrum in the majority-spin channel for (a) Si-La and (c) Ge-Lu. Right: Dependence of majority-spin anomalous Hall conductivities as a function of the Fermi-level position for (b) Si-La and (d) Ge-Lu. (All without SOC.)