  Discontinuous quantum phase transitions besides their general interest are
clearly relevant to the study of heavy fermions and magnetic transition metal
compounds. Recent results show that in many systems belonging to these classes
of materials, the magnetic transition changes from second order to first order
as they approach the quantum critical point (QCP). We investigate here some
mechanisms that may be responsible for this change. Specifically the coupling
of the order parameter to soft modes and the competition between different
types of order near the QCP. For weak first order quantum phase transitions
general results are obtained. In particular we describe the thermodynamic
behavior at this transition when it is approached from finite temperatures.
This is the discontinuous equivalent of the non-Fermi liquid trajectory close
to a conventional QCP in a heavy fermion material.
