Spin textures obtained from the Monte Carlo simulation. (a) The double-Q collinear magnetic bubble lattice. (b) The triple-Q collinear magnetic bubble lattice. (c) The triple-Q skyrmion lattice. Here the arrows represent the in-plane components and the background color (with red for positive and blue for negative) stands for the z component of local spins. (d)-(f) Electronic spectral functions as a function of φ 2 ( ˜ φ 2 ) for conduction electrons coupled to the three spin textures (a)-(c), respectively. Here the purple (yellow) denotes low (high) density of states.

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... magnets with several typical multiple-Q spin textures obtained from the Monte Carlo simulation (see Appendix E). Here we consider three kinds of spin textures: double-Q and triple-Q collinear magnetic bubble lattices where spins are nearly parallel or antiparallel due to the strong easy axis anisotropy, and triple-Q skyrmion lattice as shown in Figs. 1(a)-1(c), respectively. For conduction electrons hopping on a square lattice (whose lattice constant is set to unity), the Q ν vectors for the double-Q magnetic bubble lattice are Q 1 = (2π/5, 0) and Q 2 = (0, 2π/5), while for the triple-Q magnetic bubble and skyrmion lattices ...

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... identify the topology of electronic states in magnets with the multiple-Q spin textures, we first study their electronic spectra in the hybrid momentum space. For the double-Q magnetic bubble lattice, the translational motion of the spin texture is parameterized by φ ν = ω ν τ. The energy spectrum as a function of φ 2 (with φ 1 = 0) is shown in Figs. 1(d) for the double-Q spin texture with J = 1.5t and B = 0. Here we use the periodic boundary condition along the x direction and the open boundary condition along the y direction. Apparently, there are topological edge states in the bulk energy gaps indicating the system is topologically nontrivial. For the triple-Q spin textures, to make ...

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... and the final temperature is typically taken at T = 0.001-0.01. We also start the simulations from the multiple-Q spin configurations, such as the double-Q and triple-Q collinear spin textures. We determine the magnetic phase by comparing their energies of the obtained magnetic patterns in simulations. The double-Q collinear spin texture in Fig. 1(a) is obtained at T = 0.01, K = 0.5, α = 1.0, A = 0.5, and |Q| = 2π/6 on the square lattice, while the triple-Q collinear spin texture in Fig. 1(b) is obtained at T = 0.001, K = 1.0, α = 0.0, A = 0.0, and |Q| = 2π/6 on the triangular ...

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... as the double-Q and triple-Q collinear spin textures. We determine the magnetic phase by comparing their energies of the obtained magnetic patterns in simulations. The double-Q collinear spin texture in Fig. 1(a) is obtained at T = 0.01, K = 0.5, α = 1.0, A = 0.5, and |Q| = 2π/6 on the square lattice, while the triple-Q collinear spin texture in Fig. 1(b) is obtained at T = 0.001, K = 1.0, α = 0.0, A = 0.0, and |Q| = 2π/6 on the triangular ...

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... magnetic phases in the frustrated spin model are obtained from the Monte Carlo simulations based on the Metropolis algorithm. The lattices have N = 100 × 100 spins and the periodic boundary conditions. In the simulations, the 10 5 − 10 7 Monte Carlo sweeps measurements are performed after equilibration. The triple-Q skyrmion spin texture in Fig. 1(c) is obtained at T = 0.01, H = 0.27, and A = ...