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arXiv:1601.02632 · Physical review. B./Physical review. B

Bosonic analog of a topological Dirac semimetal: Effective theory, neighboring phases, and wire construction

Matthew F. Lapa, Gil Young Cho, Taylor L. Hughes

Published Dec 5, 2016• 4 citations
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Abstract

We construct a bosonic analog of a two-dimensional topological Dirac semimetal (DSM). The low-energy description of the most basic 2D DSM model consists of two Dirac cones at positions $\ifmmode\pm\else\textpm\fi{}{\mathbf{k}}_{0}$ in momentum space. The local stability of the Dirac cones is guaranteed by a composite symmetry ${Z}_{2}^{\mathcal{TI}}$, where $\mathcal{T}$ is time reversal and $\mathcal{I}$ is inversion. This model also exhibits interesting time-reversal and inversion symmetry breaking electromagnetic responses. In this work we construct a bosonic version by replacing each Dirac cone with a copy of the $O(4)$ nonlinear sigma model (NLSM) with topological theta term and theta angle $\ensuremath{\theta}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$. One copy of this NLSM also describes the gapless surface termination of the 3D bosonic topological insulator (BTI). We compute the time-reversal and inversion symmetry breaking electromagnetic responses for our model and show that they are twice the value one gets in the DSM case matching what one might expect from, for example, a bosonic Chern insulator. We also investigate the stability of the BSM model and find that the composite ${Z}_{2}^{\mathcal{TI}}$ symmetry again plays an important role. Along the way we clarify many aspects of the surface theory of the BTI including the electromagnetic response, the charges and statistics of vortex excitations, and the stability to symmetry-allowed perturbations. We briefly comment on the relation between the various descriptions of the $O(4)$ NLSM with $\ensuremath{\theta}=\ensuremath{\pi}$ used in this paper (a dual vortex description and a description in terms of four massless fermions) and the recently proposed dual description of the BTI surface in terms of $2+1$-dimensional quantum electrodynamics with two flavors of fermion $(N=2 {\text{QED}}_{3})$. In a set of four Appendixes we review some of the tools used in the paper and also derive some of the more technical results.

DOI
10.1103/physrevb.94.245110

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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 three dimensions 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 can not be fully realized in a purely two-dimensional system. We also confirm that the surface phases of the bosonic topological insulator proposed by Vishwanath and Senthil [Phys. Rev. X 3, 011016 (2013)] provide a consistent termination of a bulk exhibiting the statistical Witten effect. In a forthcoming paper, we will provide an explicit field-theoretic, lattice-regularized construction of the three-dimensional topological insulator of bosons, employing a parton decomposition and subsequent condensation of parton-monopole composites.