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Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect

Ashvin Vishwanath, T. Senthil

TL;DR

This work analyzes three-dimensional bosonic symmetry-protected topological phases that lack bulk topological order but host anomalous surface states. It develops bulk BF and theta-term field theories and presents two complementary surface descriptions—a projective-vortex/dual description and a network-model approach—that converge on a consistent picture of surface deconfined criticality and symmetry realization. A central result is that the bosonic TI can exhibit a quantized magnetoelectric response with $ heta=2\pi$, accompanied by surface states that cannot be trivially gapped without breaking symmetry or enabling topological order; the framework also accommodates topological paramagnets and a putative E$_8$-type phase with half-quantized surface thermal Hall effect. Together, these insights extend the classification and physical understanding of interacting 3D SPT phases and their exotic surface phenomena.

Abstract

We discuss physical properties of `integer' topological phases of bosons in D=3+1 dimensions, protected by internal symmetries like time reversal and/or charge conservation. These phases invoke interactions in a fundamental way but do not possess topological order and are bosonic analogs of free fermion topological insulators and superconductors. While a formal cohomology based classification of such states was recently discovered, their physical properties remain mysterious. Here we develop a field theoretic description of several of these states and show that they possess unusual surface states, which if gapped, must either break the underlying symmetry, or develop topological order. In the latter case, symmetries are implemented in a way that is forbidden in a strictly two dimensional theory. While this is the usual fate of the surface states, exotic gapless states can also be realized. For example, tuning parameters can naturally lead to a deconfined quantum critical point or, in other situations, a fully symmetric vortex metal phase. We discuss cases where the topological phases are characterized by quantized magnetoelectric response θ, which, somewhat surprisingly, is an odd multiple of 2π. Two different surface theories are shown to capture these phenomena - the first is a nonlinear sigma model with a topological term. The second invokes vortices on the surface that transform under a projective representation of the symmetry group. A bulk field theory consistent with these properties is identified, which is a multicomponent `BF' theory supplemented, crucially, with a topological term. A possible topological phase characterized by the thermal analog of the magnetoelectric effect is also discussed.

Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect

TL;DR

This work analyzes three-dimensional bosonic symmetry-protected topological phases that lack bulk topological order but host anomalous surface states. It develops bulk BF and theta-term field theories and presents two complementary surface descriptions—a projective-vortex/dual description and a network-model approach—that converge on a consistent picture of surface deconfined criticality and symmetry realization. A central result is that the bosonic TI can exhibit a quantized magnetoelectric response with , accompanied by surface states that cannot be trivially gapped without breaking symmetry or enabling topological order; the framework also accommodates topological paramagnets and a putative E-type phase with half-quantized surface thermal Hall effect. Together, these insights extend the classification and physical understanding of interacting 3D SPT phases and their exotic surface phenomena.

Abstract

We discuss physical properties of `integer' topological phases of bosons in D=3+1 dimensions, protected by internal symmetries like time reversal and/or charge conservation. These phases invoke interactions in a fundamental way but do not possess topological order and are bosonic analogs of free fermion topological insulators and superconductors. While a formal cohomology based classification of such states was recently discovered, their physical properties remain mysterious. Here we develop a field theoretic description of several of these states and show that they possess unusual surface states, which if gapped, must either break the underlying symmetry, or develop topological order. In the latter case, symmetries are implemented in a way that is forbidden in a strictly two dimensional theory. While this is the usual fate of the surface states, exotic gapless states can also be realized. For example, tuning parameters can naturally lead to a deconfined quantum critical point or, in other situations, a fully symmetric vortex metal phase. We discuss cases where the topological phases are characterized by quantized magnetoelectric response θ, which, somewhat surprisingly, is an odd multiple of 2π. Two different surface theories are shown to capture these phenomena - the first is a nonlinear sigma model with a topological term. The second invokes vortices on the surface that transform under a projective representation of the symmetry group. A bulk field theory consistent with these properties is identified, which is a multicomponent `BF' theory supplemented, crucially, with a topological term. A possible topological phase characterized by the thermal analog of the magnetoelectric effect is also discussed.

Paper Structure

This paper contains 22 sections, 92 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Transport Properties of 3D bosonic topological insulators: General Constraints
  4. Surface Theory of 3D Bosonic SPT Phases
  5. Surface States and Projective Vortices
  6. Surface States of a Bosonic topological insulator: Symmetry U(1)$\rtimes\ Z^T_2$
  7. Network Model Construction
  8. Synthesis of the Two Approaches
  9. Analysis of the Surface Field Theory: Phase Diagram, Deconfined Criticality and Fermionic Vortices
  10. 3D Topological Field Theories
  11. Two component BF theory of Bosonic Topological Insulator
  12. Bulk sigma model theory
  13. Other symmetries: Topological Paramagnets
  14. Symmetry $Z_2^T$
  15. Symmetry U(1)$\times\ Z^T_2$
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Schematic depiction of a 3D symmetry protected topological phase, with two conserved species of bosons (U(1)$\times$U(1)) and time reversal symmetry ($Z_2^T$). The bulk is insulating and corresponds to a condensate of vortices of both species (shown as black and white loops). In the topological phase, the vortex line of one species that ends on a surface carries half charge of the other species. Such a surface, it may be argued, does not have a trivial gapped phase, where the symmetries are preserved. Pictorially, the vortex lines may be viewed in 1D Haldane phases, with half charged end states.
  • Figure 2: Network model for the fermion IQHE transition. On the top all chiral edge modes are paired to yield an ordinary insulator. On the bottom there is an unpaired edge mode to yield an integer quantum Hall insulator
  • Figure 3: Network model for the boson IQHE transition. Each edge channel now has a charged chiral mode and a counter-propagating neutral mode. The rest is the same as for fermions.