It is well known that one signature of the three-dimensional electron topological insulator is the Witten effect: if the system is coupled to a compact electromagnetic gauge field, a monopole in the bulk acquires a half-odd-integer polarization charge. In the present work, we propose a corresponding phenomenon for the topological insulator of bosons in three dimensions protected by particle number conservation and time-reversal symmetry. We claim that although a monopole inside a topological insulator of bosons can remain electrically neutral, its statistics are transmuted from bosonic to fermionic. We demonstrate that this ``statistical Witten effect'' directly implies that if the surface of the topological insulator is neither gapless, nor spontaneously breaks the symmetry, it necessarily supports an intrinsic two-dimensional topological order. Moreover, the surface properties can not be fully realized in a purely two-dimensional system. We also confirm that the surface phases of the bosonic topological insulator proposed by Vishwanath and Senthil [Phys. Rev. X 3, 011016 (2013)] provide a consistent termination of a bulk exhibiting the statistical Witten effect. In a forthcoming paper, we will provide an explicit field-theoretic, lattice-regularized construction of the three-dimensional topological insulator of bosons, employing a parton decomposition and subsequent condensation of parton-monopole composites.
Topological insulators are characterized by the presence of gapless surface modes protected by time-reversal symmetry. In three space dimensions the magnetoelectric response is described in terms of a bulk $\ensuremath{\theta}$ term for the electromagnetic field. Here we construct theoretical examples of such phases that cannot be smoothly connected to any band insulator. Such correlated topological insulators admit the possibility of fractional magnetoelectric response described by fractional $\ensuremath{\theta}/\ensuremath{\pi}$. We show that fractional $\ensuremath{\theta}/\ensuremath{\pi}$ is only possible in a gapped time-reversal-invariant system of bosons or fermions if the system also has deconfined fractional excitations and associated degenerate ground states on topologically nontrivial spaces. We illustrate this result with a concrete example of a time-reversal-symmetric topological insulator of correlated bosons with $\ensuremath{\theta}=\frac{\ensuremath{\pi}}{4}$. Extensions to electronic fractional topological insulators are briefly described.
We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators (TBIs) to interacting many-particle systems known as symmetry-protected topological (SPT) phases. As with the TBIs, these states have a bulk gap and no exotic excitations but have nontrivial surface states that are protected by symmetry. We describe the various possible phases and their properties in three-dimensional systems with realistic symmetries. We develop many key ideas for the theory of these states using simple examples. The emphasis is on physical rather than mathematical properties. We survey insights obtained from the study of SPT phases for a number of other theoretical problems.
Symmetry protected topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper, we describe the classification and properties of such phases for three-dimensional (3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free-fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases, we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal
U(1) subgroup we show that this classification is complete. We describe the nontrivial surface and bulk properties of these states. In particular, we discuss interesting correlated surface states that are not captured in a free-fermion description. We show that in many, but not all, cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.
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.
The surfaces of three-dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time-reversal
T or
U(1) charge conservation symmetry. Here, we discuss a possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then, the surface must develop topological order of a kind that can not be realized in a two-dimensional (2D) system with the same symmetries. We discuss candidate surface states, non-Abelian quantum Hall states which, when realized in 2D, have ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ and hence break
T symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as
T-symmetric surface states. The corresponding 3D phases are confined, and have $\ensuremath{\theta}=\ensuremath{\pi}$ magnetoelectric response. Two candidate states have the same 12-particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their
T transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24-particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a
T-symmetric surface state.