Considerable efforts have been made in recent years to theoretically understand quantum phase transitions in Kondo lattice systems. A particular focus is on Kondo destruction, which leads to quantum criticality that goes beyond the Landau framework of order-parameter fl uctuations. This unconventional quantum criticality has provided an understanding of the unusual dynamical scaling observed experimentally. It also predicted a sudden jump of the Fermi surface and an extra (Kondo destruction) energy scale, both of which have been veri fi ed by systematic experiments. Considerations of Kondo destruction have in addition yielded a global phase diagram, which has motivated the current interest in heavy fermion materials with variable dimensionality or geometrical frustration. Here we summarize these developments, and discuss some of the ongoing work and open issues. We also consider the implications of these results for superconductivity. Finally, we address the effect of spin – orbit coupling on the global phase diagram, suggest that SmB 6 under pressure may display unconventional superconductivity in the transition regime between a Kondo insulator phase and an antiferroamgnetic metal phase, and argue that the interfaces of heavy-fermion heterostructures will provide a fertile setting to explore topological properties of both Kondo insulators and heavy- fermion superconductors

Interest in the heavy fermion metals has motivated us to examine the quantum phases and their Fermi surfaces within the Kondo lattice model. We demonstrate that the model is soluble asymptotically exactly in any dimension d>1, when the Kondo coupling is small compared with the RKKY interaction and in the presence of antiferromagnetic ordering. We show that the Kondo coupling is exactly marginal in the renormalization group sense, establishing the stability of an ordered phase with a small Fermi surface AFS. Our results have implications for the global phase diagram of the heavy fermion metals, suggesting a Lifshitz transition inside the antiferromagnetic region and providing a new perspective for a Kondo-destroying antiferromagnetic quantum critical point.

Metallic magnetism is both ancient and modern, occurring in such familiar settings as the lodestone in compass needles and the hard drive in computers. Surprisingly, a rigorous theoretical basis for metallic ferromagnetism is still largely missing. The Stoner approach perturbatively treats Coulomb interactions when the latter need to be large, whereas the Nagaoka approach incorporates thermodynamically negligible holes into a half-filled band. Here, we show that the ferromagnetic order of the Kondo lattice is amenable to an asymptotically exact analysis over a range of interaction parameters. In this ferromagnetic phase, the conduction electrons and local moments are strongly coupled but the Fermi surface does not enclose the latter (i.e., it is "small"). Moreover, non-Fermi-liquid behavior appears over a range of frequencies and temperatures. Our results provide the basis to understand some long-standing puzzles in the ferromagnetic heavy fermion metals, and raise the prospect for a new class of ferromagnetic quantum phase transitions. Fermi surface | itinerant magnetism | non-Fermi liquidA contemporary theme in quantum condensed matter physics concerns competing ground states and the accompanying novel excitations (1). With a plethora of different phases, magnetic heavy fermion materials should reign supreme as the prototype for competing order. So far, most of the theoretical scrutiny has focused on antiferromagnetic heavy fermions (2, 3). Nonetheless, the list of heavy fermion metals that are known to exhibit ferromagnetic order continues to grow. An early example subjected to extensive studies is CeRu 2 Ge 2 (ref. 4 and references therein). Other ferromagnetic heavy fermion metals include CePt (5), CeSi x (6), CeAgSb 2 (7), and URu 2−x Re x Si 2 at x > 0.15 (8, 9). More recently discovered materials include CeRuPO (10) and UIr 2 Zn 20 (11). Finally, systems such as UGe 2 (12) and URhGe (13) are particularly interesting because they exhibit a superconducting dome as their metallic ferromagnetism is tuned toward its border. Some fascinating and general questions have emerged (14,15,16), yet they have hardly been addressed theoretically. One central issue concerns the nature of the Fermi surface: Is it "large," encompassing both the local moments and conduction electrons as in paramagnetic heavy fermion metals (17, 18), or is it "small," incorporating only conduction electrons? Measurements of the de Haas-van Alphen (dHvA) effect have suggested that the Fermi surface is small in CeRu 2 Ge 2 (14-16), and have provided evidence for Fermi surface reconstruction as a function of pressure in UGe 2 (19,20). At the same time, it is traditional to consider the heavy fermion ferromagnets as having a large Fermi surface when their relationship with unconventional superconductivity is discussed (12,13,21); an alternative form of the Fermi surface in the ordered state could give rise to a new type of superconductivity near its phase boundary. All these point to the importance of theoretically understanding the ferromagnet...

We discuss the general theoretical arguments advanced earlier for the T = 0 global phase diagram of antiferromagnetic Kondo lattice systems, distinguishing between the established and the conjectured. In addition to the wellknown phase of a paramagnetic metal with a "large" Fermi surface (P L ), there is also an antiferromagnetic phase with a "small" Fermi surface (AF S ). We provide the details of the derivation of a quantum non-linear sigma-model (QNLσ M) representation of the Kondo lattice Hamiltonian, which leads to an effective field theory containing both low-energy fermions in the vicinity of a Fermi surface and low-energy bosons near zero momentum. An asymptotically exact analysis of this effective field theory is made possible through the development of a renormalization group procedure for mixed fermion-boson systems. Considerations on how to connect the AF S and P L phases lead to a global phase diagram, which not only puts into perspective the theory of local quantum criticality for antiferromagnetic heavy fermion metals, but also provides the basis to understand the surprising recent experiments in chemically-doped as well as pressurized YbRh 2 Si 2 . We point out that the AF S phase still occurs for the case of an equal number of spin-1/2 local moments and conduction electrons. This observation raises the prospect for a global phase diagram of heavy fermion systems in the Kondo-insulator regime. Finally, we discuss the connection between the Kondo breakdown physics discussed