Effective Crystalline Electric Field Potential in a j-j Coupling Scheme

We propose an effective model on the basis of a $j$-$j$ coupling scheme to describe local $f$-electron states for realistic values of Coulomb interaction $U$ and spin-orbit coupling $\lambda$, for future development of microscopic theory of magnetism and superconductivity in $f^n$-electron systems, where $n$ is the number of local $f$ electrons. The effective model is systematically constructed by including the effect of a crystalline electric field (CEF) potential in the perturbation expansion in terms of $1/\lambda$. In this paper, we collect all the terms up to the first order of $1/\lambda$. Solving the effective model, we show the results of the CEF states for each case of $n$=2$\sim$5 with $O_{\rm h}$ symmetry in comparison with those of the Stevens Hamiltonian for the weak CEF. In particular, we carefully discuss the CEF energy levels in an intermediate coupling region with $\lambda/U$ in the order of 0.1 corresponding to actual $f$-electron materials between the $LS$ and $j$-$j$ coupling schemes. Note that the relevant energy scale of $U$ is the Hund's rule interaction. It is found that the CEF energy levels in the intermediate coupling region can be quantitatively reproduced by our modified $j$-$j$ coupling scheme, when we correctly take into account the corrections in the order of $1/\lambda$ in addition to the CEF terms and Coulomb interactions which remain in the limit of $\lambda$=$\infty$. As an application of the modified $j$-$j$ coupling scheme, we discuss the CEF energy levels of filled skutterudites with $T_{\rm h}$ symmetry.Comment: 12 pages, 7 figures. Typeset with jpsj2.cl

Multipole State of Heavy Lanthanide Filled Skutterudites

We discuss multipole properties of filled skutterudites containing heavy lanthanide Ln from a microscopic viewpoint on the basis of a seven-orbital Anderson model. For Ln=Gd, in contrast to naive expectation, quadrupole moments remain in addition to main dipole ones. For Ln=Ho, we find an exotic state governed by octupole moment. For Ln=Tb and Tm, no significant multipole moments appear at low temperatures, while for Ln=Dy, Er, and Yb, dipole and higher-order multipoles are dominant. We briefly discuss possible relevance of these multipole states with actual materials.Comment: 5 pages, 3 figure

Multipole as ff-Electron Spin-Charge Density in Filled Skutterudites

It is shown that $f$-electron multipole is naturally defined as spin-charge one-electron density operator in the second-quantized form with the use of tensor operator on the analogy of multipole expansion of electromagnetic potential from charge distribution in electromagnetism. Due to this definition of multipole, it is possible to determine multipole state from a microscopic viewpoint on the basis of the standard linear response theory for multipole susceptibility. In order to discuss multipole properties of filled skutterudites, we analyze a seven-orbital impurity Anderson model by employing a numerical renormalization group method. We show our results on possible multipole states of filled skutterudite compounds.Comment: To appear in the Proceedings of International Conference on "New Quantum Phenomena in Skutterudite and Related Systems" (September 2007, Kobe, Japan

Microscopic Approach to Magnetism and Superconductivity of ff-Electron Systems with Filled Skutterudite Structure

In order to gain a deep insight into $f$-electron properties of filled skutterudite compounds from a microscopic viewpoint, we investigate the multiorbital Anderson model including Coulomb interactions, spin-orbit coupling, and crystalline electric field effect. For each case of $n$=1$\sim$13, where $n$ is the number of $f$ electrons per rare-earth ion, the model is analyzed by using the numerical renormalization group (NRG) method to evaluate magnetic susceptibility and entropy of $f$ electron. In order to make further step to construct a simplified model which can be treated even in a periodic system, we also analyze the Anderson model constructed based on the $j$-$j$ coupling scheme by using the NRG method. Then, we construct an orbital degenerate Hubbard model based on the $j$-$j$ coupling scheme to investigate the mechanism of superconductivity of filled skutterudites. In the 2-site model, we carefully evaluate the superconducting pair susceptibility for the case of $n$=2 and find that the susceptibility for off-site Cooper pair is clearly enhanced only in a transition region in which the singlet and triplet ground states are interchanged.Comment: 14 pages, 11 figures, Typeset with jpsj2.cl

Orbital ordering phenomena in dd- and ff-electron systems

In recent decades, novel magnetism of $d$- and $f$-electron compounds has been discussed very intensively both in experimental and theoretical research fields of condensed matter physics. It has been recognized that those material groups are in the same category of strongly correlated electron systems, while the low-energy physics of $d$- and $f$-electron compounds has been separately investigated rather in different manners. One of common features of both $d$- and $f$-electron systems is certainly the existence of active orbital degree of freedom, but in $f$-electron materials, due to the strong spin-orbit interaction in rare-earth and actinide ions, the physics seems to be quite different from that of $d$-electron systems. In general, when the number of internal degrees of freedom and relevant interactions is increased, it is possible to obtain rich phase diagram including large varieties of magnetic phases by using several kinds of theoretical techniques. However, we should not be simply satisfied with the reproduction of rich phase diagram. It is believed that more essential point is to seek for a simple principle penetrating complicated phenomena in common with $d$- and $f$-electron materials, which opens the door to a new stage in orbital physics. In this sense, it is considered to be an important task of this article to explain common features of magnetism in $d$- and $f$-electron systems from a microscopic viewpoint, using a key concept of orbital ordering, in addition to the review of the complex phase diagram of each material group.Comment: 112 pages, 38 figure

Construction of a Microscopic Model for Yb and Tm Compounds on the Basis of a \mib{j}-\mib{j} Coupling Scheme

We provide a prescription to construct a microscopic model for heavy lanthanide systems such as Yb and Tm compounds by exploiting a $j$-$j$ coupling scheme. Here we consider a situation with a large spin-orbit coupling, in which $j$=5/2 sextet is fully occupied, while $j$=7/2 octet is partially occupied, where $j$ denotes total angular momentum. We evaluate crystalline electric field potentials and Coulomb interactions among the states of the $j$=7/2 octet to construct a local Hamiltonian in the $j$-$j$ coupling scheme. Then, it is found that the local $f$-electron states composed of the $j$=7/2 octet agree quite well with those of seven $f$ orbitals even for a realistic value of the spin-orbit coupling. As an example of the application of the present model, we discuss low-temperature multipole states of Yb- and Tm-based filled skutterudites by analyzing multipole susceptibility of the Anderson model in the $j$-$j$ coupling scheme with the use of a numerical renormalization group technique. From the comparison with the numerical results of the seven-orbital Anderson model, it is concluded that the multipole state is also well reproduced by the $j$-$j$ coupling model, even when we include the hybridization between conduction and $f$ electrons for the realistic value of the spin-orbit coupling. Finally, we briefly discuss future applications of the present prescription for theoretical research on heavy lanthanide compounds.Comment: 12 pages, 8 figures

Exciton Mediated Triplet Superconductivity in Th System PrOs4Sb12

In PrOs4Sb12, the lowest-lying singlet and triplet states in a Pr 4f^2 configuration hybridize with conduction electrons having local a_u and t_u point-group symmetries. It is shown that for an attractive triplet pairing interaction, the orbital degrees of freedom of the t_u component are important. In addition, the Th point-group symmetry characteristic of skutterudites plays an important role in stabilizing triplet superconductivity.Comment: 4 pages, 2 figure