# Achieving Bosonic Topological Insulators by Condensing Vortex-lines of Superfluids

As a bosonic analog of topological insulators, bosonic topological insulators (BTI) in three spatial dimensions are symmetry-protected topological phases (SPT) with U(1)$\rtimes$Z$^T_2$ symmetry, where U(1) is boson particle number conservation, and Z$^T_2$ is time-reversal symmetry with $\mathcal{T}^2=1$. Such kinds of new states of matter were firstly proposed based on the group cohomology classification theory'' which states there are two distinct root states.  Each  root state carries a $\mathbb{Z}_2$ index. Soon after, their corresponding surface anomalous topological orders were proposed, which even leads to a new BTI state, i.e. a new root state that is beyond group cohomology classification.     Nevertheless, it is still unclear what is the universal physical mechanism for BTI phases and what kinds of microscopic Hamiltonian can realize them. To answer the first question may generically shed light on the second question.  In this work, we propose a universal physical mechanism via vortex-line condensation in a superfluid, e.g. helium-4. Using such a simple physical picture, we find three  BTI root states, in which two of them are classified by group cohomology theory while the rest is beyond group cohomology classification. The physical picture also leads to a natural'' bulk dynamic topological quantum field theory (TQFT) description for BTI phases and gives rise to a physical way of thinking towards experimental realizations. Finally, we generalize the vortex-line condensation picture into other symmetries and find that in three dimensions, even for a unitary Z$_2$ symmetry, there is a nontrivial Z$_2$ SPT phase beyond the group cohomology classification. Some related references: arXiv:1410.2594, arXiv:1212.2121.

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