1d Lattice Models for the Boundary of 2d “Majorana” Fermion SPTs: Kramers-Wannier Duality as an Exact Z₂ Symmetry.

A symmetry protected topological phase (SPT) is one which cannot be smoothly deformed into a trivial phase without breaking the symmetry. The symmetry which protects the phase acts "anomalously" on the edge of the SPT - for instance, in a non-onsite manner. Here, we examine the case when the anomaly is even more severe and the effective edge Hilbert space of the SPT does not admit a local tensor-product structure.

We begin with Fidkowski and Tarantino's lattice model for the root SPT phase for the symmetry G = Z₂ x Z₂^f in 2d in the presence of an edge. Each site in the edge carries an Ising spin, and each domain wall hosts a Majorana fermion, so the size of the boundary Hilbert space does not scale simply with the number of sites. We introduce a description of this Hilbert space akin to the Jordan-Wigner bosonization of a fermion chain and find the action of the Z₂ symmetry on it. We construct a 3-site Hamiltonian which respects the Z₂ symmetry and find strong numerical evidence that it realizes an Ising CFT where the Z₂ symmetry acts as Kramers-Wannier duality.