One-dimensional lattice models for the boundary of two-dimensional Majorana fermion symmetry-protected topological phases: Kramers-Wannier duality as an exact Z2 symmetry

It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of (2+1)-dimensional [(2+1)D] fermion SPTs. We focus on the root SPT phase for the symmetry group G = Z(2) x Z(2)(f). Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the Z(2) symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the Z(2) symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the (2+1)D topological superconductor protected by time-reversal symmetry T with T-2 = (-1)(F).

Automated summary

This paper investigates the boundary problem of root SPT phase with symmetry group G = Z(2) x Z(2)(f) in (2+1)D fermion SPTs. By processing the bulk model, it derives a one-dimensional lattice model for the boundary and finds that it realizes an Ising conformal field theory with a stable gapless boundary state.