UC Santa Barbara

As condensed matter theorists, we always try to seek new quantum phases of matter that are not possible in classical physics. In this dissertation, I discussed a new type of quantum disordered phases known as symmetry protected topological (SPT) phases, which is a generalization of the topological insulator to interacting fermion or boson/spin systems with various symmetries. In the first part of this thesis, a nonlinear σ-model (NLσM) field theory is introduced as a powerful tool to describe the properties of the bosonic SPT phases. Secondly, we want to answer the question of how to detect the SPT states from their bulk properties. Introducing gauge fields was shown to be an effective theoretical tool to study bulk properties of SPT phases. Furthermore, we investigated anyon and loop statistics of gauged SPT states in the framework of NLσM. We also designed a new numerical probe, so-called strange correlator, which can distinguish SPT states from trivial states based on the bulk wavefunction on a closed manifold. Thirdly, several aspects of surface states of SPT phases are discussed. 1. A surface phase transition of 3d topological insulator is studied through a new controlled expansion method with the help of the recently discovered fermion-vortex duality. 2. A new strongly interacting conformal field theory on the surface of 3d bosonic SPT state is also found by a controlled renormalization group calculation. 3. we made a connection between the surface of SPT phase and the Lieb-Schultz-Mattis (LSM) theorem, which enables us to identity the SU(N) and SO(N) spin systems that permit a featureless ground state in 2d and 3d. Finally, we proposed the first experimental realization of bosonic SPT state in dimension higher than 1. We established a general relation between interacting multi-layer fermionic SPTs and bosonic SPT with the same symmetry, which motivates a proposal of realizing 2+1d bosonic SPT phase in bilayer graphene system.