Unwinding Short-Range Entanglement

TL;DR

This work demonstrates that short-range-entangled SPT phases in 1+1D can be unwound to trivial states by extending the global symmetry to a larger group $\tilde{G}$ and applying symmetry-invariant finite-depth unitaries, in line with the symmetry-extension framework of Wang–Wen–Witten. It provides explicit constructive protocols for both bosonic and fermionic SPTs, including AKLT/Haldane, cluster, and layered Kitaev-chain models, and connects the unwinding procedure to Schur-cover theory and group cohomology. The results show that symmetry-extension can reduce or eliminate SPT classifications (e.g., CII from $\mathbb{Z}_2$ to trivial; AIII from $\mathbb{Z}_4$ to $\mathbb{Z}_2$; BDI from $\mathbb{Z}_8$ to $\mathbb{Z}_4$), while noting limitations for inherently fermionic cases and odd-$\nu$ Kitaev chains. The work unifies symmetry-breaking, inversion, and symmetry-extension as routes to unwind SRE, and suggests broader implications for boundary anomalies, fixed-point constructions, and potential connections to gravity/cobordism frameworks.

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.

Figures (25)

• Figure 1: $H_1$, which belongs to the trivial phase in the space of Hamiltonians without symmetry, can become non-trivial in the space of Hamiltonians with some symmetry $G$ .
• Figure 2: A finite depth unitary circuit (FDUC).
• Figure 3: Unwinding the Haldane phase by explicitly breaking symmetry. (FDUC).
• Figure 4: Trivializing the Haldane phase by symmetry extension.
• Figure 5: A representative SPT state.