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Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators

Peng Ye, Zheng-Cheng Gu

TL;DR

The paper develops a physical, hydrodynamic route to three-dimensional bosonic topological insulators (BTIs) via vortex-line condensation in a 3D superfluid, deriving a bulk BF+BB TQFT that captures all three BTI root states (including one beyond group cohomology) and rigorously connects them to surface topological orders. By analyzing gauge structure, quantization, and symmetry implementations, the authors classify trivial and nontrivial BTIs through GL$(N,\mathbb{Z})$ and extended GL transformations, and demonstrate a unique nontrivial BTI arising from an $\Lambda_{so8}$ block. They also present two pure $b\wedge d a$ constructions that realize BTIs with unusual symmetry definitions, showing a many-to-one correspondence between surface orders (e.g., Z$_p$ toric codes) and the bulk, and extend the framework to Z$_N$ SPTs beyond group cohomology. Overall, the work provides a concrete, physically intuitive mechanism for BTI realization, clarifies bulk–boundary correspondence, and suggests broader beyond-cohomology SPTs in 3D.

Abstract

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation {symmetries}. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a $\mathbb{Z}_2$ index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via \textit{vortex-line condensation} {from} a 3d superfluid to achieve all {three} root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are \textit{rigorously} derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to $Z_N$ symmetries and discuss potential SPT phases beyond the group cohomology classification.

Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators

TL;DR

The paper develops a physical, hydrodynamic route to three-dimensional bosonic topological insulators (BTIs) via vortex-line condensation in a 3D superfluid, deriving a bulk BF+BB TQFT that captures all three BTI root states (including one beyond group cohomology) and rigorously connects them to surface topological orders. By analyzing gauge structure, quantization, and symmetry implementations, the authors classify trivial and nontrivial BTIs through GL and extended GL transformations, and demonstrate a unique nontrivial BTI arising from an block. They also present two pure constructions that realize BTIs with unusual symmetry definitions, showing a many-to-one correspondence between surface orders (e.g., Z toric codes) and the bulk, and extend the framework to Z SPTs beyond group cohomology. Overall, the work provides a concrete, physically intuitive mechanism for BTI realization, clarifies bulk–boundary correspondence, and suggests broader beyond-cohomology SPTs in 3D.

Abstract

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation {symmetries}. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via \textit{vortex-line condensation} {from} a 3d superfluid to achieve all {three} root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are \textit{rigorously} derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to symmetries and discuss potential SPT phases beyond the group cohomology classification.

Paper Structure

This paper contains 28 sections, 85 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Hydrodynamical approach to topological quantum field theory
  4. 3d superfluid state and its dual description
  5. Trivial Mott insulators realized by condensing vortex-lines (strings)
  6. Adding a vortex-line (string) linking Berry phase term into the trivial Mott insulator
  7. General properties of the topological quantum field theory
  8. Gauge transformation and Bulk $\mathbb{GL}(N,\mathbb{Z})$ transformation
  9. Quantization conditions on path-integral field variables
  10. Bosonic topological insulators in the presence of $b\wedge b$ topological term
  11. Definition of time-reversal transformation
  12. Quantization condition implemented by time-reversal symmetry
  13. Trivial SPT states with $|\text{det}\Lambda|=1$
  14. Trivial SPT states with $|\text{det}\Lambda|>1$
  15. Extended $\mathbb{GL}$ transformation and nontrivial SPT states
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Phases obtained by vortex-line condensation. In the Phase transition-1, U(1) symmetry (i.e. boson number conservation) is restored from superfluid to a trivial Mott insulator by condensing strings (i.e. 2$\pi$-vortex-lines). Thus, the trivial Mott insulator phase is formed by vortex-line condensation with $b\wedge \mathrm{d}a$ type bulk field theory description. In the Phase transition-2, strings are also condensed and the bulk field theory is also $b\wedge \mathrm{d}a$ type (see Sec. \ref{['sec:pbf2']}, \ref{['sec:pbf1']}). But the resultant Mott phase is a nontrivial SPT state (i.e. bosononic topological insulators, BTI) since either U(1) or Z$^T_2$ symmetry transformations is defined in an unusual way. Thus, we end up with two different BTI root states. In the Phase transition-3, strings are condensed in the presence of a nontrivial linking Berry phase term, or more precisely, an nontrivial multicomponent $b\wedge b$ type term. The nontrivial Mott phase is a BTI phase obtained in Sec. \ref{['sec:btifti']}, which is a SPT root state beyond group cohomology classification and supports "all-fermion" Z$_2$ surface topological order. Here, Z$^T_2$ denotes time-reversal symmetry with $\mathcal{T}^2=1$.
  • Figure 2: Physical meaning of $b\wedge b$ topological term. The larger loop denotes a vortex-line that is static and located on $xy$-plane. The smaller loop is perpendicular to $xy$-plane, parallel to $yz$-plane, and moves toward $z$-direction. There are four snap-shots shown in this figure from left to right. The blue dot in the third snap-shot denotes the intersection of two loops. In the fourth snap-shot, two loops are eventually linked to each other. $b\wedge b$ term will contribute a phase at the third snap-shot of the unlinking-linking process.

Theorems & Definitions (3)

  • Conjecture 1
  • Definition 1
  • Definition 2