Hedgehog lattice and field-induced chirality in breathing-pyrochlore Heisenberg antiferromagnets

We theoretically investigate a ${J}₁\text{\ensuremath{-}}{J}₃$ classical Heisenberg model on the breathing pyrochlore lattice, where the nearest-neighbor (NN) exchange interactions for small and large tetrahedra, ${J}₁$ and ${J}₁^{\ensuremath{'}}$, take different values due to the breathing bond alternation and ${J}₃$ is the third NN antiferromagnetic interaction along the bond direction. It is found by means of Monte Carlo simulations that for large ${J}₃$, a hedgehog lattice, a three-dimensional periodic array of magnetic monopoles and antimonopoles, emerges in the form of a quadruple-$\mathbf{Q}$ state characterized by the ordering vector of $\mathbf{Q}=(\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{1}{2})$, being irrespective of the signs of ${J}₁$ and/or ${J}₁^{\ensuremath{'}}$ as long as ${J}₁\ensuremath{\ne}{J}₁^{\ensuremath{'}}$. It is also found that in an applied magnetic field, there appear six quadruple-$\mathbf{Q}$ states depending on the values of ${J}₁$ and ${J}₁^{\ensuremath{'}}$, among which three phases including the in-field hedgehog-lattice state exhibit nonzero total chirality ${\mathbit{\ensuremath{\chi}}}^{\mathrm{T}}$ associated with the anomalous Hall effect of chirality origin. In the remaining two chiral phases, which are realized in the presence of ferromagnetic ${J}₁$ and/or ${J}₁^{\ensuremath{'}}$, the spin structure is not topologically nontrivial, in spite of the fact that ${\mathbit{\ensuremath{\chi}}}^{\mathrm{T}}\ensuremath{\ne}0$. The role of the topological objects of the monopoles in ${\mathbit{\ensuremath{\chi}}}^{\mathrm{T}}$ is also discussed.