Unwinding Short-Range Entanglement

TL;DR

This paper demonstrates that certain short-range entangled symmetry-protected topological phases can be transformed into trivial phases by enlarging the symmetry group, effectively unwinding their topological properties.

Contribution

It introduces a method to unwind non-trivial SPT phases through symmetry extension, expanding the understanding of topological phase manipulation.

Findings

Unwinding of bosonic and fermionic SPT phases in 1+1 dimensions.

Symmetry extension can lift zero mode degeneracy.

Applicable to any 1+1 D bosonic SPT with finite on-site symmetry.

Abstract

Symmetry-Protected Topological (SPT) phases are gapped phases of quantum matter protected by global symmetries that cannot be adiabatically deformed to a trivial phase without breaking symmetry. In this work, we show that, for several SPT phases that are short range entangled (SRE), enlarging symmetries may effectively achieve the consequences of explicitly breaking symmetries. In other words, we demonstrate that non-trivial SPT phases can be unwound to trivial ones by symmetry extension- through a path where the Hilbert space is enlarged and the Hamiltonian is invariant under an extended symmetry group applying the idea of Wang, Wen and Witten in arXiv:1705.06728. We show examples of both bosonic and fermionic SPT phases in 1+1 dimensions, including Haldane's bosonic spin chain and layers of Kitaev's fermionic Majorana chains. By adding degrees of freedom into the boundary/bulk, we can lift the zero mode degeneracy, or unwind the whole system. Furthermore, based on properties of Schur cover, we sketch a general picture of unwinding applicable to any 1+1 D bosonic SPT phase protected by on-site finite symmetry. Altogether we show that SRE states can be unwound by symmetry breaking, inversion and symmetry extension.