We investigate the role of quantum fluctuations in topological quantum phase transitions of quantum spin-Hall insulators and quantum anomalous-Hall insulators. Employing the variational cluster approximation to obtain the single-particle Green's function of the interacting many-body system, we characterize different phases by direct calculation of the recently proposed topological order parameter for interacting systems. We pinpoint the influence of quantum fluctuations on the quantum spin Hall to Mott insulator transition in several models. Furthermore, we propose a general mechanism by which a topological quantum phase transition can be driven by the divergence of the self-energy induced by interactions.
A comprehensive theory of the quantum phase transition in clean itinerant Heisenberg ferromagnets is presented. It is shown that the standard mean-field description of the transition is invalid in spatial dimensions
d< 3 due to the existence of soft particle-hole excitations that couple to the order parameter fluctuations and lead to an upper critical dimension
dc+=3. A generalized mean-field theory that takes these additional modes into account predicts a fluctuation-induced first-order transition. In a certain parameter regime, this first-order transition in turn is unstable with respect to a fluctuation-induced second-order transition. The quantum ferromagnetic transition may thus be either of first or of second order, in agreement with experimental observations. A detailed discussion is given of the stability of the first-order transition and of the critical behavior at the fluctuation-induced second-order transition. In
d=3, the latter is mean field like with logarithmic corrections to scaling, and in
d<3 it can be controlled by means of a $3\ensuremath{-}\ensuremath{\epsilon}$ expansion.
We introduce a coherence susceptibility method, based on the fact that it signals quantum fluctuations, for identifying quantum phase transitions, which are induced by quantum fluctuations. This method requires no prior knowledge of order parameter, and there is no need for careful considerations concerning the choice of a bipartition of the system. It can identify different types of quantum phase transition points exactly. At finite temperatures, where quantum criticality is influenced by thermal fluctuations, our method can pinpoint the temperature frame of quantum criticality, which perfectly coincides with recent experiments.