Field-induced multiple-Q phases in a frustrated quantum magnet

We study a frustrated hard-core boson model inspired by recent experiments on the field-induced quantum phase transition in the S$=$1 dimer antiferromagnet Ba$_{3}$Mn$_{2}$O$_{8}$ for a magnetic field H parallel to the c axis. We calculate the effective interactions in the low-density limit by adding the ladder diagrams and determine the ground state phase diagram near the quantum critical point. The phase diagram is very rich and includes different multiple-\textbf{Q} Bose-Einstein condensates (BECs) that combine the six degenerate incommensurate lowest-energy modes $\pm$ \textbf{Q}$_{n}$ (1$\le n\le $3) at the quantum critical point. The multiple-\textbf{Q} states include a lattice of magnetic vortices that emerges out of frustration between the boson-boson interactions.