Stripe melting, a transition between weak and strong symmetry protected topological phases

摘要

For a gapped disordered many-body system with both internal and translationsymmetry, one can define the corresponding weak and strong Symmetry ProtectedTopological (SPT) phases. A strong SPT phase is protected by the internalsymmetry $G$ only while a weak SPT phase, fabricated by alignment of strong SPTstate in a lower dimension, requires additional discrete translation symmetryprotection. In this paper, we construct a phase transition between weak andstrong SPT phase in strongly interacting boson system. The starting point ofour construction is the superconducting Dirac fermions with pair densitywave(PDW) order in $2d$. We first demonstrate that the nodal line of the PDWcontains a $1d$ boson SPT phase. We further show that melting the PDW stripeand condensing the nodal line provoke the transition from weak to strong SPTphase in $2d$. The phase transition theory contains an O(4)non-linear-$\sigma$-model(NL$\sigma$M) with topological $\Theta$-term emergingfrom the proliferation of domain walls bound to an SPT chain. Similar schemealso applies to weak-strong SPT transition in other dimensions and predictspossible phase transition from $2d$ to $3d$ topological order.