Frustration, induced by competition between different interactions, can be realized on a diamond lattice, especially in A-site of spinel materials with a chemical formula AB2X 4. We address the realization of classical spin liquid and provide a unified theoretical explanation of various phenomena observed in spinel materials.

Motivated by CoNb2O6, we study interplay between quantum criticality and geometric frustration. In the presence of carefully oriented magnetic field, the system realizes quantum phase transition of transverse field quantum Ising model while magnetic frustration exists due to the lattice geometry at the same time. We reveal several quantum magnetic phases and crossovers between one dimensional quantum criticality and several three dimensional quantum critical points.

Lastly, we look for the possible realization of a quantum spin liquid focusing on A-site magnets in rare-earth pyrochlores. Based on a pseudospin-1/2 Hamiltonian, we discuss how quantum fluctuations lead the exotic phases such as a U(1) quantum spin liquid and XY ordered phases, beyond classical spin ice state.