  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.
