First order quantum phase transitions
  Quantum phase transitions have been the subject of intense investigations in the last two decades [1]. Among other problems, these phase transitions are relevant in the study of heavy fermion systems, high temperature superconductors and Bose-Einstein condensates. More recently there is increasing evidence that in many systems which are close to a quantum critical point (QCP) different phases are in competition. In this paper we show that the main effect of this competition is to give rise to inhomogeneous behavior associated with quantum first order transitions. These effects are described theoretically using an action that takes into account the competition between different order parameters. The method of the effective potential is used to calculate the quantum corrections to the classical functional. These corrections generally change the nature of the QCP and give rise to interesting effects even in the presence of non-critical fluctuations. An unexpected result is the appearance of an inhomogeneous phase with two values of the order parameter separated by a first order transition. Finally, we discuss the universal behavior of systems with a weak first order zero temperature transition in particular as the transition point is approached from finite temperatures. The thermodynamic behavior along this line is obtained and shown to present universal features. 
M. A. Continentino and A. S. Ferreira