Phase transitions and new phases of matter have continued to challenge our understanding of the quantum condensed matter systems. For decades, phase transitions were thought to be facilitated by the breaking of some symmetry. For example, magnetic ordering emerges due to breaking of the time-reversal symmetry; the crystallization of water into ice, on the other hand, involves broken translational symmetry. This paradigm, however, was challenged with the discovery of topological order in some condensed matter systems in the year 1982. Since then many electronic phases with novel topologies have been predicted theoretically and few have also been realized experimentally. Examples of these electronic phases, across the research frontier, span from graphene to the topological insulators and beyond. Still, several theoretical predictions of various novel topological phases are yet to be realized experimentally. Due to this reason, significant efforts are concentrated around materials with strong spin-orbit interaction, which provide a fertile ground for experimental realization of novel topological phases.
Much of condensed matter is about how different kinds of order emerge from inter- actions between many simple constituents. An ideal example of this is the family of transition metal oxides (TMOs) where a plethora of novel phenomena such as high-TC superconductivity, colossal magnetoresistance, and metal-insulator transitions could be observed as a result of the complex interplay of various electronic interactions such as electron correlations, crystal field splitting and spin-orbit coupling. The magnitude of these interactions depends on a number of factors like the atomic number of the transition metal, underlying crystal geometry, ligand environment around the transition metal, etc. Understanding the interplay of these interactions which ultimately decides the ground state is one of the fundamental problems in condensed matter physics.
Failure of band theory to explain the insulating ground state in TMOs highlighted the importance of electron-electron interactions. In TMOs, the conduction is driven by d- orbitals which are spatially compact when compared to the s- or p-orbitals in simple metals. This results in strong Coulomb repulsion between the d electrons of TMOs. The transition from metallic to insulating state due to strong electron correlations was successfully modelled theoretically via Hubbard model given by John Hubbard.
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