Author(s):AtMa P. O. Chan

While many features of topological band insulators are commonly discussed at

the level of single-particle electron wave functions, such as the gapless Dirac

spectrum at their boundary, it remains elusive to develop a {\it hydrodynamic}

or {\it 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, and

the implication of its duality, of the surface of topological insulators are

also discussed. We also touch upon the hydrodynamic theory of fractional

topological insulators by using the parton construction.

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