Inhomogeneous Kondo-lattice in geometrically frustrated Pr(2)Ir(2)O(7)

Magnetic fluctuations induced by geometric frustration of local Ir-spins disturb the formation of long-range magnetic order in the family of pyrochlore iridates. As a consequence, Pr(2)Ir(2)O(7) lies at a tuning-free antiferromagnetic-to-paramagnetic quantum critical point and exhibits an array of complex phenomena including the Kondo effect, biquadratic band structure, and metallic spin liquid. Using spectroscopic imaging with the scanning tunneling microscope, complemented with machine learning, density functional theory and theoretical modeling, we probe the local electronic states in Pr(2)Ir(2)O(7) and find an electronic phase separation. Nanoscale regions with a well-defined Kondo resonance are interweaved with a non-magnetic metallic phase with Kondo-destruction. These spatial nanoscale patterns display a fractal geometry with power-law behavior extended over two decades, consistent with being in proximity to a critical point. Our discovery reveals a nanoscale tuning route, viz. using a spatial variation of the electronic potential as a means of adjusting the balance between Kondo entanglement and geometric frustration.