Topological phases and novel transport phenomena in two-dimensional quantum systems

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Recently, a new method to study fractional quantum Hall effect (FQHE) has been proposed: we impose torus boundary conditions to the two-dimensional electron systems, and map the system to a one-dimensional lattice model. Furthermore, the electron-electron interactions are simplified considering the thin-torus limit. Using this method, we have discussed the relationship between Haldane conjecture in quantum spin systems and FQHE. We also proposed a model with an exact ground state describing the FQH state, and matrix product expression for the wave function.We also discussed interlayer electric conduction, and dimerized state in a quantum spin chain with three-body interactions.