  A Valence bond solid (VBS) is a nonmagnetic, long-range ordered state of a
quantum spin system where local spin singlets are formed in some regular
pattern. We here study the competition between VBS order and a fully polarized
ferromagnetic state as function of an external magnetic field in a
one-dimensional extended Heisenberg model---the J-Q$_2$ model---using
stochastic series expansion quantum Monte Carlo simulations with directed loop
updates. We discuss the ground state phase diagram.
