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Bosonic topological insulator in three dimensions and the statistical Witten effect

Max A. Metlitski, C. L. Kane, Matthew P. A. Fisher

TL;DR

The paper identifies a bulk diagnostic for the 3D bosonic topological insulator protected by $U(1)\ltimes Z_2^T$: the statistical Witten effect, where monopole statistics become fermionic for a nontrivial theta angle, is periodic only modulo $4\pi$. This bulk phenomenon implies that any TR-preserving, gapped surface must host intrinsic 2D topological order, and cannot be realized in a purely 2D system. The work connects this bulk anomaly to Vishwanath–Senthil surface phases and situates the bosonic TI within the cohomology classification, while promising an explicit lattice construction in a companion paper.

Abstract

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 3d 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 cannot be fully realized in a purely 2d system. We also confirm that the surface phases of the bosonic topological insulator proposed by Vishwanath and Senthil (arXiv:1209.3058) provide a consistent termination of a bulk exhibiting the statistical Witten effect. In a companion paper, we will provide an explicit field-theoretic, lattice-regularized, construction of the 3d topological insulator of bosons, employing a parton decomposition and subsequent condensation of parton-monopole composites.

Bosonic topological insulator in three dimensions and the statistical Witten effect

TL;DR

The paper identifies a bulk diagnostic for the 3D bosonic topological insulator protected by : the statistical Witten effect, where monopole statistics become fermionic for a nontrivial theta angle, is periodic only modulo . This bulk phenomenon implies that any TR-preserving, gapped surface must host intrinsic 2D topological order, and cannot be realized in a purely 2D system. The work connects this bulk anomaly to Vishwanath–Senthil surface phases and situates the bosonic TI within the cohomology classification, while promising an explicit lattice construction in a companion paper.

Abstract

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 3d 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 cannot be fully realized in a purely 2d system. We also confirm that the surface phases of the bosonic topological insulator proposed by Vishwanath and Senthil (arXiv:1209.3058) provide a consistent termination of a bulk exhibiting the statistical Witten effect. In a companion paper, we will provide an explicit field-theoretic, lattice-regularized, construction of the 3d topological insulator of bosons, employing a parton decomposition and subsequent condensation of parton-monopole composites.

Paper Structure

This paper contains 12 sections, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Witten effect and fermionic topological insulators
  3. The statistical Witten effect and the bosonic topological insulator
  4. Surface of the bosonic topological insulator and the statistical anomaly
  5. Surface phases with $\sigma_{xy} = \pm 1$.
  6. Surface superfluid.
  7. Surface phase with $Z_2$ topological order.
  8. General constraints on the surface.
  9. Relation to the cohomology classification
  10. Conclusion
  11. Statistical interactions near the $\sigma_{xy} = 1$ surface of a bosonic topological insulator.
  12. Statistics of superfluid vortices on the surface of a bosonic topological insulator.

Figures (1)

  • Figure 1: Electric charge $q$ of dyons with magnetic flux $2 \pi$ as a function of the $\theta$-angle in a bosonic insulator. Red lines denote dyons with bosonic statistics and blue lines - dyons with fermionic statistics. Although the allowed values of electric charge are invariant under $\theta \to \theta + 2 \pi$, the corresponding statistics is periodic only modulo $4\pi$.