Magnets with noncentrosymmetric lattice structures can host a three-dimensional noncoplanar spin texture called the magnetic hedgehog lattice (HL) with a periodic array of magnetic monopoles and anti-monopoles. Despite recent discovery of two types of short-period HLs in MnSi1−xGex, their microscopic origin remains elusive. Here, we study the stability of such magnetic HLs for an effective spin model with long-range interactions arising from the itinerant nature of electrons. By variational calculations and simulated annealing, we find that the HLs are stabilized in the ground state at zero magnetic field by the synergetic effect of the antisymmetric exchange interactions generated by the spin-orbit coupling and the multiple-spin interactions generated by the spin-charge coupling. We also clarify the full phase diagram in the magnetic field, which includes multiple phase transitions with changes in the number of monopoles and anti-monopoles.Chirality, often termed as handedness, is a key concept in a broad field of science, ranging from particle physics to biology. In condensed matter physics, chiral magnetic textures, which break both inversion and mirror symmetries in additon to time-reversal symmetry, have recently attracted considerable attention for potential applications to next-generation electronic devices. There are a variety of the chiral magnetic textures, such as skyrmion lattices [1] and chiral soliton lattices [2]. Noncollinear and noncoplanar spin arrangements in these textures generate emergent electromagnetic fields through the Berry phase mechanism, which induce unconventional transport, optical, and magnetoelectric properties [3][4][5].Recently, a three-dimensional chiral magnetic texture, which is called the hedgehog lattice (HL), was discovered in the B20-type compound MnGe [6,7]. The magnetic structure is characterized by cubic three wave vectors, and hence, it is referred as the triple-Q hedgehog lattice (3Q-HL) [ Fig. 1(a)]. The 3Q-HL has a periodic array of hyperbolic hedgehog and anti-hedgehog spin textures, which generates an emergent magnetic field with a periodic array of radial hedgehogs and anti-hedgehogs regarded as magnetic monopoles and anti-monopoles, as shown in Fig. 1(c) [8][9][10]. The peculiar magnetic field was discussed as a source of the enormous topological Hall effect [11] and thermoelectoric effect [12,13]. In addition, in MnSi 1−x Ge x , the 3Q-HL changes into a different HL characterized by tetrahedral four wave vectors, dubbed the quadruple-Q hedgehog lattice (4Q-HL) [ Fig. 1(b)] [14,15]. Remarkably, the magnetic periods of these 3Q-and 4Q-HLs are very short ∼ 2-3 nm, in contrast to most of the skyrmion lattices.Such magnetic HLs have been theoretically studied prior to the experimental discovery, e.g., by the Ginzburg-Landau theory [16], variational calculations [17], and Monte Carlo (MC) simulations [18]. The variational study for a classical spin model showed that the 3Q-HL is not stabilized, whereas the 4Q-HL is obtained in an applied magnetic field [17]. The 4Q...
