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arXiv:1502.07752 · Physical Review B

Topological orders with global gauge anomalies

Yi‐Zhuang You, Cenke Xu

Published Aug 6, 2015• 6 citations
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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 (conjecture) 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 $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the $3d$ spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a $4d$ topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ 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. Our study also leads to exotic states of matter in pure $3d$ space.

DOI
10.1103/physrevb.92.054410

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We show how 1+11+11+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of 1+1d1+1\mathrm{d}1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.