While many features of topological band insulators are commonly discussed at the level of single-particle electron wave functions, such as the gapless Dirac boundary spectrum, it remains elusive to develop a hydrodynamic or collective description of fermionic topological band insulators in 3+1 dimensions. As the Chern-Simons theory for the 2+1-dimensional quantum Hall effect, such a hydrodynamic effective field theory provides a universal description of topological band insulators, even in the presence of interactions, and that of putative fractional topological insulators. In this paper, we undertake this task by using the functional bosonization. The effective field theory in the functional bosonization is written in terms of a two-form gauge field, which couples to a
U(1) gauge field that arises by gauging the continuous symmetry of the target system [the
U(1) particle number conservation]. Integrating over the
U(1) gauge field by using the electromagnetic duality, the resulting theory describes topological band insulators as a condensation phase of the
U(1) gauge theory (or as a monopole condensation phase of the dual gauge field). The hydrodynamic description of the surface of topological insulators and the implication of its duality are also discussed. We also touch upon the hydrodynamic theory of fractional topological insulators by using the parton construction.
Effective field theories that describe the dynamics of a conserved U(1) current in terms of ``hydrodynamic'' degrees of freedom of topological phases in condensed matter are discussed in general dimension
D=d+1 using the functional bosonization technique. For noninteracting topological insulators (superconductors) with a conserved U(1) charge and characterized by an integer topological invariant [more specifically, they are topological insulators in the complex symmetry classes (class A and AIII), and in the ``primary series'' of topological insulators, in the eight real symmetry classes], we derive the BF-type topological field theories supplemented with the Chern-Simons (when
D is odd) or the $\ensuremath{\theta}$ (when
D is even) terms. For topological insulators characterized by a
Z2 topological invariant (the first and second descendants of the primary series), their topological field theories are obtained by dimensional reduction. Building on this effective field theory description for noninteracting topological phases, we also discuss, following the spirit of the parton construction of the fractional quantum Hall effect by Block and Wen, the putative ``fractional'' topological insulators and their possible effective field theories, and use them to determine the physical properties of these nontrivial quantum phases.
We study the structure of the ground-state wave functions of bosonic symmetry protected topological (SPT) insulators in three space dimensions. We demonstrate that the differences with conventional insulators are captured simply in a dual vortex description. As an example, we show that a previously studied bosonic topological insulator with both global U(1) and time-reversal symmetry can be described by a rather simple wave function written in terms of dual ``vortex ribbons.'' The wave function is a superposition of all the vortex-ribbon configurations of the boson, and a factor $(\ensuremath{-}1)$ is associated with each self-linking of the vortex ribbons. This wave function can be conveniently derived using an effective field theory of the SPT phase in the strong-coupling limit, and it naturally explains all the phenomena of this SPT phase discussed previously. The ground-state structure for other three-dimensional (3D) bosonic SPT phases are also discussed similarly in terms of vortex loop gas wave functions. We show that our methods reproduce known results on the ground-state structure of some 2D SPT phases.