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318 Handbook on the Physics and Chemistry of Rare Earths In addition, later it was found that standard HM theory develops some anomalies in two dimensions that render it incorrect. This originates from the fact that antiferromagnetic fluctuations (z1⁄42) in two dimensions have effective dimension d+z1⁄44, exactly the upper critical dimension where interactions become marginally relevant. It has been found that one cannot integrate out fermions to arrive at an effective model, as Hertz originally did. Instead, both electrons and critical fluctuations need to be treated at an equal level, which is an ongoing theoretical effort (Abanov and Chubukov, 2000, 2004; Lee, 2009; Metlitski and Sachdev, 2010; Pankov et al., 2004; Varma, 2015). 3.4 Critical Quasiparticles While Kondo hybridization certainly takes on an important role in many heavy fermion compounds, an alternative approach to the Kondo breakdown picture is to describe the effective mass divergence phenomenologically. The quasiparticle mass enhancement is characterized by the parameter Z1⁄4m/m*, and hence experimentally we find Z!0 at the transition. This would naively imply that the Fermi surface and hence the materials metallic behavior would disappear. Yet, the way around this is via a “critical” Fermi surface (Senthil, 2008) where Z1⁄40 but the spectral function still displays some kind of nonanalytic behavior. One can write down phenomenologically the Greens function for the corresponding critical quasiparticles (Abrahams et al., 2014; Vojta et al., 2015; W€olfle and Abrahams, 2011, 2015), from which many critical exponents can be extracted that match experimental results. Nonetheless, a microscopic picture of why quasiparticles would become critical is still lacking. 3.5 Conclusion The theoretical field seems to be in a state of confusion. At the one hand there are microscopic models based on Kondo physics between f- and c-electrons, but those seem to be nearly impossible to solve rigorously. On the other hand, it is possible to describe a number of experiments by purely phenomenologi- cal models. It remains a challenge for the theorists to find a good microscopic basis for their phenomenology. In addition, new experimental results such as ARPES at low temperatures, are then necessary to distinguish between differ- ent scenarios of quantum criticality beyond HM. 4 PRESENT MATERIALS AND EXPERIMENTS There are numerous lanthanide-based heavy-fermion liquids that are claimed to exhibit QPT’s. Rather than attempting to catalogue and evaluate these many different materials and their quantum criticality claims, we offer here some overview articles where the reader can find these intermetallicPDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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