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Composite electronic orders induced by orbital Kondo effect

In a large number of rare-earth and actinide systems, Kondo effect tends to suppress magnetic order by making the spin singlet between localized and conduction electron spins. In the presence of orbital degrees of freedom, however, there emerge exotic electronic orders induced by Kondo effect. The orbital Kondo effect can collectively make diagonal and off-diagonal (superconducting) orders. With the particle-hole symmetry in conduction bands, these orders are all degenerate, forming a macroscopic SO(5) multiplet. This paper discusses recent theoretical development on these electronic orders which are relevant to Pr$^{3+}$ and U$^{4+}$ systems with even number of $f$ electrons per site. In the superconducting order, each conduction-electron pair is coupled with local degrees of freedom, forming a composite entity with a staggered spatial pattern. The quasi-particle spectrum is best interpreted as virtual hybridization with resonant states at the Fermi level. Possible order parameter for URu$_2$Si$_2$ in the hidden order state is discussed in the context of composite orders. Briefly discussed are related issues such as homogeneous odd-frequency pairing and SO(5) theory for high-temperature superconductors.