Topological Orders with Global Gauge Anomalies

Abstract

By definition, the physics of the $d-$dimensional (dim) boundary of a $(d+1)-$dim symmetry protected topological (SPT) state cannot be realized as itself on a $d-$dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a $d-$dim theory must be a boundary or not, is to couple this theory to a gauge field (or to "gauge" its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue that if the gauge anomaly of the boundary is "perturbative", then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ge 2$. We will demonstrate this conclusion with two examples: (1) the $3d$ spin-1/2 chiral fermion with the well-known Witten's global anomaly, which is the boundary of a $4d$ topological superconductor with SU(2) or U(1)$⋊Z_2$ symmetry; and (2) the $4d$ boundary of a $5d$ topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped ${\mathbb{Z}}_{2N}$ topological order with topological degeneracy, but this ${\mathbb{Z}}_{2N}$ topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects.

12 pages

References in corpus (3)

• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Anomalies of discrete symmetries in three dimensions and group cohomology

Cited by in corpus (5)

• Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions
• Realizing topological surface states in a lower-dimensional flat band
• Decorated defect condensate, a window to unconventional quantum phase transitions in Weyl semimetals
• Impact of lepton flavour universality violation on CP violation sensitivity of long baseline neutrino oscillation experiments
• Competing Orders and Anomalies