Study on Heavy Fermion in Eu-based Compounds by the Numerical Renormalization Group Method

In this paper, we report the theoretical results regarding unconventional heavy fermion (HF) states realizable in Europium (Eu) systems and related materials. Many rare-earth and actinide-based compounds exhibit magnetic ordering at low temperatures. In particular, the electron state of Gd and Eu systems is characterized by a large pure spin S=7 /2, and thus magnetic ordering becomes quite stable. However, recently, a HF behavior has been observed in Eu systems, implying a different mechanism from the conventional HF formation. It is also interesting that the HF is realized in the intermediate valence state in the Eu system. Many Eu systems indicate the valence fluctuation between Eu2+ and Eu3+. In the case of divalent, as mentioned above, it is difficult for the HF state to surpass magnetic order. Also, when the c-f hybridization process is restricted, the 4f spin cannot be completely screened out, and a non-magnetic state does not form. On the other hand, in the case of trivalent, the ground state of 4f electrons is a non-magnetic singlet state, and thus the Kondo effect does not occur. Therefore, the Kondo effect in the intermediate valence state of the Eu ion is highly nontrivial. To investigate the Kondo effect and the valence fluctuation in the Eu system, we introduced the effective Eu-based impurity Anderson model (Eu-IAM) considering the valence fluctuations of Eu systems and analyzed it by using numerical renormalization groups. From the results, it was clarified that a non-magnetic ground state is always formed. Furthermore, it was quantitatively clarified that the formation of nonmagnetic states deeply involves small spin-orbit coupling which is unique to Eu systems. Next, in order to analyze the HF state confirmed by Eu-IAM more closely, we derived an effective model in the vicinity of Eu2+ state, which is the S = 7/2 underscreening (US) Kondo model with a crystal field△S/2 derived from valence fluctuations. The NRG results show that the ground state exhibits a strong coupling fixed point similar to the usual Kondo singlet. However, it was found that reducing the crystal field remarkably increases the effective mass, and furthermore, applying a magnetic field also causes a mass enhancement, revealing properties contrary to normal HF states. We consider that such anomalous behavior can be caused by the ferromagnetic Kondo coupling which occurs between the US 4f state and the conduction electron. Based on this consideration, we analyzed the renormalized parameters of a weakly anisotropic ferromagnetic Kondo model. Then, we proposed a hierarchical Kondo effect as the origin of the formation of the nontrivial ground state.