It is well known that mean-field theories fail to reproduce the experimentally known critical exponents. The traditional argument which explain this failure of mean-field theories near a critical point is the Ginsburg criterion in which diverging fluctuations of the order parameter is the root cause. We argue, contrary to the above mentioned traditional view, that diverging fluctuations in real physical systems near a critical point are genuine consequence of the breakdown of the property of statistical independence, and are faithfully reproduced by the mean-field theory. By looking at the problem from the point of view of "statistical independence" the divergence of fluctuations in real physical systems near criticality becomes immediately apparent as a connection can be established between diverging correlation length and diverging fluctuations. To address the question of why mean-field theories, much successful qualitatively, fail to reproduce the known values of critical indices we argue, using the essential ideas of the Wilsonian renormalization group, that mean-field theories fail to capture the long length scale averages of an order parameter near a critical point.
The fluctuations-driven continuous quantum criticality has sparked tremendous interest in condensed matter physics. It has been verified that the fluctuations of gapless fermions can change the nature of phase transition at criticality. In this paper, we study the fermionic quantum criticality with enlarged Ising
×Ising fluctuations in honeycomb lattice materials. The perturbative renormalization group approach is employed to investigated the flow equations and stable fixed-point structure. By including cubic terms of order parameter, we found that the phase transition is always continuous for any finite value of the number of Dirac fermions
Nf. Further, we also compute the critical exponents and predict the critical scaling behavior for
Nf=2 case which corresponds to spin-1/2 fermions on graphene-like materials.
When an isolated quantum system is driven out of equilibrium, expectation values of general observables start oscillating in time. This paper reviews the general theory of such temporal fluctuations. We first survey some results on the strength of such temporal fluctuations. For example temporal fluctuations are exponentially small in the system's volume for generic systems whereas they fall-off algebraically in integrable systems. We then concentrate on the so-called quench scenario where the system is driven out-of-equilibrium under the application of a sudden perturbation. For sufficiently small perturbations, temporal fluctuations of physical observables can be characterized in full generality and can be used as an effective tool to probe quantum criticality of the underlying model. In the off-critical region the distribution becomes Gaussian. Close to criticality the distribution becomes a universal function uniquely characterized by a single critical exponent, that we compute explicitly. This contrasts standard equilibrium quantum fluctuations for which the critical distribution depends on a numerable set of critical coefficients and is known only for limited examples. The possibility of using temporal fluctuations to determine pseudo-critical boundaries in optical lattice experiments is further reviewed.