From second to first order transitions in a disordered quantum magnet

We study the spin-glass transition in a disordered quantum model. There is a region in the phase diagram where quantum effects are small and the phase transition is second order, as in the classical case. In another region, quantum fluctuations drive the transition first order. Across the first order line the susceptibility is discontinuous and shows hysteresis. Our findings reproduce qualitatively observations on LiHo$_x$Y$_{1-x}$F$_4$. We also discuss a marginally stable spin-glass state and derive some results previously obtained from the real-time dynamics of the model coupled to a bath.