Field-induced Multi-$\boldsymbol{\vec{Q}}$ States in a Pyrochlore Heisenberg Magnet

We construct exact ground states of the $J_1$-$J_{3b}$ classical Heisenberg model on the pyrochlore lattice in the presence of a magnetic field. They are non-coplanar multi-$\vec{Q}$ spin configurations with a large magnetic unit cell that generalize the previously found coplanar sublattice pairing states. Using linear spin wave theory, we show that entropy favors these multi-$\vec{Q}$ states at low temperatures in high magnetic fields. This is confirmed by Monte Carlo simulations, and a phase diagram is constructed. We also calculate the zero-temperature dynamical structure factor. Besides the usual Goldstone modes associated with the ordering $\vec{Q}$s, we find high intensity gapless modes at momenta where there are no Bragg peaks.