Rare regions, i.e., rare large spatial disorder fluctuations, can dramatically change the properties of a phase transition in a quenched disordered system. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. Stronger effects can be observed at zero-temperature quantum phase transitions, at nonequilibrium phase transitions, and in systems with correlated disorder. In some cases, rare regions can actually completely destroy the sharp phase transition by smearing.This topical review presents a unifying framework for rare region effects at weakly disordered classical, quantum, and nonequilibrium phase transitions based on the effective dimensionality of the rare regions. Explicit examples include disordered classical Ising and Heisenberg models, insulating and metallic random quantum magnets, and the disordered contact process.

This review discusses a paradigm that has become of increasing importance in the theory of quantum phase transitions, namely, the coupling of the order-parameter fluctuations to other soft modes and the resulting impossibility of constructing a simple Landau-Ginzburg-Wilson theory in terms of the order parameter only. The soft modes in question are manifestations of generic scale invariance, i.e., the appearance of long-range order in whole regions in the phase diagram. The concept of generic scale invariance and its influence on critical behavior is explained using various examples, both classical and quantum mechanical. The peculiarities of quantum phase transitions are discussed, with emphasis on the fact that they are more susceptible to the effects of generic scale invariance than their classical counterparts. Explicit examples include the quantum ferromagnetic transition in metals, with or without quenched disorder; the metal-superconductor transition at zero temperature; and the quantum antiferromagnetic transition. Analogies with classical phase transitions in liquid crystals and classical fluids are pointed out, and a unifying conceptual framework is developed for all transitions that are influenced by generic scale invariance.

It is shown that the phase transition in low-T c clean itinerant ferromagnets is generically of first order, due to correlation effects that lead to a nonanalytic term in the free energy. A tricritical point separates the line of first order transitions from Heisenberg critical behavior at higher temperatures. Sufficiently strong quenched disorder suppresses the first order transition via the appearance of a critical end point. A semiquantitative discussion is given in terms of recent experiments on MnSi, and predictions for other experiments are made. [S0031-9007(99)09305-9] PACS numbers: 75.20.En, 64.60.Kw, 75.45. + j The thermal paramagnet-to-ferromagnet transition at the Curie temperature T C is usually regarded as a prime example of a second order phase transition. For materials with high T C this is well established both experimentally and theoretically. Recently there has been a considerable interest in the corresponding quantum phase transition of itinerant electrons at zero temperature (T 0), and in the related finite T properties of weak itinerant ferromagnets, i.e., systems with a very low T C . Experimentally, the transition in the weak ferromagnet MnSi has been tuned to different T C by applying hydrostatic pressure [1]. Interestingly, the transition at low T was found to be of first order, while at higher transition temperatures it is of second order [2]. The tricritical temperature that separates the two types of transitions was found to roughly coincide with the location of a maximum in the magnetic susceptibility in the paramagnetic phase. Theoretically, it has been shown [3,4] that in a T 0 itinerant electron system, soft modes that are unrelated to the critical order parameter (OP) or magnetization fluctuations couple to the latter. This leads to an effective long-range interaction between the OP fluctuations. In disordered systems, the additional soft modes are the same "diffusons" that cause the so-called weak-localization effects in paramagnetic metals [5]. In clean systems there are analogous, albeit weaker, effects that manifest themselves as corrections to Fermi liquid theory [6]. A Gaussian theory is sufficient to obtain the exact quantum critical behavior in the most interesting dimension, d 3, for clean as well as for disordered systems (apart from logarithmic corrections in the clean case) [3,4].In this Letter, we show that at sufficiently low temperatures the phase transition in itinerant ferromagnets is generically of first order. This surprising result is shown to be rooted in fundamental and universal many-body physics underlying the transition, viz. long-wavelength correlation effects, and, hence to be independent of the band structure. This suggests that the behavior observed in MnSi is generic, and should also be present in other weak itinerant ferromagnets. We also make detailed predictions about how quenched disorder suppresses the first order transition, which allows for decisive experimental checks of our theory.Let us start by deriving the functional form of the free en...

The wavevector and temperature dependent static spin susceptibility, \chi_s(Q,T), of clean interacting Fermi systems is considered in dimensions 1\leq d \leq 3. We show that at zero temperature \chi_s is a nonanalytic function of |Q|, with the leading nonanalyticity being |Q|^{d-1} for 1