  The mineral linarite, PbCuSO$_4$(OH)$_2$, is a spin 1/2 chain with
frustrating nearest neighbor ferromagnetic and next-nearest neighbor
antiferromagnetic exchange interactions. Our inelastic neutron scattering
experiments performed above the saturation field establish that the ratio
between these exchanges is such that linarite is extremely close to the quantum
critical point between spin-multipolar phases and the ferromagnetic state.
However, the measured complex magnetic phase diagram depends strongly on the
magnetic field direction. The field-dependent phase sequence is explained by
our classical simulations of a nearly critical model with tiny orthorhombic
exchange anisotropy. The simulations also capture qualitatively the measured
variations of the wave vector as well as the staggered and the uniform
magnetizations in an applied field.
