| Many fascinating and striking physical phenomena in condensed matter physics, including the fractional quantum Hall effect and high temperature superconductivity, are believed to be due to the effects of strong correlations between electrons. However, a significant portion of strongly correlated phenomena are qualitatively different from those in weakly interacting systems, and are at best only partially understood. Consequently, the formal construction of strongly correlated models with exactly solvable or asymptotically solvable limits can give, in a controlled way, some insight concerning such fascinating phenomena as fractionalized excitations in an insulator and superconductivity from only repulsive interactions.;Competing orders and exotic phases are believed to occur generically in strongly correlated systems. In the second part, we study the checkerboard Hubbard model, which consists of weakly coupled square plaquettes. We find that there are at least 17 phases, including a d-Mott insulator, a d-wave but nodeless superconductor, and a nematic phase. We further study the checkerboard Hubbard model on a finite lattice by numerical exact diagonalization and find evidence that superconductivity is optimized by intermediate inhomogeneity (inter-plaquette hopping about half of the intra-plaquette hopping) and intermediate interaction strength (U comparable to the bandwidth). We argue that this leads to some insights relevant to the cuprate phenomenology.;In the final part, we study the nematic order and quantum magnetism in the recently discovered iron based superconductors. It is argued that the lattice distortion observed in neutron scattering experiments originates from the nematic ordering of the spin degrees of freedom on Fe atoms. Consequently, both the lattice distortion and quantum magnetism can be understood in a unified picture.;This thesis is divided into three parts. In the first part, exactly or asymptotically solvable models are introduced to formally demonstrate the existence of three types of fractionalized insulators as a proof of principle: a chiral spin liquid with non-Abelian anyon excitations, an algebraic spin liquid with gapless "spinon" excitations or a pseudo "spinon" Fermi surface, and a gapped Z2 spin liquid with genuine spin-charge separation. |
