Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality

Abstract

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.

23 pages, 5 figures; References added; Paper updated to match published version

References in corpus (7)

• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Characterizing topological order by studying the ground states of an infinite cylinder
• Entanglement, combinatorics and finite-size effects in spin-chains
• Critical entanglement spectrum of one-dimensional symmetry protected topological phases
• Entanglement Spectra of the quantum hard-square model: Holographic minimal models

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