Multi-scale Quantum Criticality driven by Kondo-lattice Coupling in Pyrochlore Systems

Pyrochlore systems ($A_{2}B_{2}O_{7}$) with $A$-site rare-earth local moments and $B$-site $5d$ conduction electrons offer excellent material platforms for the discovery of exotic quantum many-body ground states. Notable examples include U(1) quantum spin liquid of the local moments and semimetallic non-Fermi liquid of the conduction electrons. Here we investigate emergent quantum phases and their transitions driven by the Kondo lattice coupling between such highly entangled quantum ground states. Using the renormalization group method, it is shown that weak Kondo lattice coupling is irrelevant, leading to a fractionalized semimetal phase with decoupled local moments and conduction electrons. Upon increasing the Kondo lattice coupling, this phase is unstable to the formation of broken symmetry states. Particularly important is the opposing influence of the Kondo lattice coupling and long-range Coulomb interaction. The former prefers to break the particle-hole symmetry while the latter tends to restore it. The characteristic competition leads to possibly multiple phase transitions, first from a fractionalized semimetal phase to a fractionalized Fermi surface state with particle-hole pockets, followed by the second transition to a fractionalized ferromagnetic state. Multiscale quantum critical behaviors appear at nonzero temperatures and with external magnetic field near such quantum phase transitions. We discuss the implication of these results to the experiments on Pr$_2$Ir$_2$O$_7$.