Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality
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In a system with chiral topological order, there is a remarkable
correspondence between the edge and entanglement spectra: the low-energy
spectrum of the system in the presence of a physical edge coincides with the
lowest part of the entanglement spectrum (ES) across a virtual cut of the
system, up to rescaling and shifting. In this paper, we explore whether the
edge-ES correspondence extends to nonchiral topological phases. Specifically,
we consider the Wen-plaquette model which has Z_2 topological order. The
unperturbed model displays an exact correspondence: both the edge and
entanglement spectra within each topological sector a (a = 1,...,4) are flat
and equally degenerate. Here, we show, through a detailed microscopic
calculation, that in the presence of generic local perturbations: (i) the
effective degrees of freedom for both the physical edge and the entanglement
cut consist of a spin-1/2 chain, with effective Hamiltonians H_edge^a and
H_ent.^a, respectively, both of which have a Z_2 symmetry enforced by the bulk
topological order; (ii) there is in general no match between their low energy
spectra, that is, there is no edge-ES correspondence. However, if supplement
the Z_2 topological order with a global symmetry (translational invariance
along the edge/cut), i.e. by considering the Wen-plaquette model as a symmetry
enriched topological phase (SET), then there is a finite domain in Hamiltonian
space in which both H_edge^a and H_ent.^a realize the critical Ising model,
whose low-energy effective theory is the c = 1/2 Ising CFT. This is achieved
because the presence of the global symmetry implies that both Hamiltonians, in
addition to being Z_2 symmetric, are Kramers-Wannier self-dual. Thus, the bulk
topological order and the global translational symmetry of the Wen-plaquette
model as a SET imply an edge-ES correspondence at least in some finite domain
in Hamiltonian space.